The algebra of semi-flows: a tale of two topologies

The algebra of semi-flows: a tale of two topologies

Kelly Spendlove        Robert Vandervorst

K. Spendlove Mathematical Institute

University of Oxford,  Oxford UK

R.C.A.M. Vandervorst Department of Mathematics

VU University Amsterdam, The Netherlands

To capture the global structure of a dynamical system  we reformulate dynamics in terms of appropriately constructed

topologies, which we call flow topologies; we call this process topologization.
This yields a description of a semi-flow in terms of a bi-topological space, with the first topology corresponding to the (phase) space and the second to the flow topology.

A study of topology is facilitated  through discretization, i.e. defining and examining appropriate finite sub-structures. Topologizing the dynamics provides an elegant solution to their discretization by discretizing the associated flow topologies.

We introduce Morse pre-orders, an instance of a more general bi-topological discretization, which synthesize the space and flow topologies, and encode the directionality of  dynamics.
We describe how
Morse pre-orders can be augmented with appropriate (co)homological information in order to describe invariance of the dynamics; this ensemble provides an algebraization of the semi-flow.
An illustration of the main ingredients of the paper is provided
by an application to
the theory of discrete parabolic flows.
Algebraization  yields a new invariant for positive braids in terms of a bi-graded differential module which  contains Morse theoretic information of  parabolic flows.

06-04 06-08 06D05 06E05 37B35 37M22 37C70

§ PRELUDE

We introduce the point of view that dynamics can be studied as a topology.  Thus an analysis of a dynamical system results in an analysis of two topologies (i.e. a bi-toplogical space): the first topology corresponds to the (phase) space, and the second to the dynamics.

Topology and associated algebraic invariants have long played a prolific role in the theory of dynamical systems [13, 48, 50, 54].  Loosely stated, a dynamical system engenders topological data, both local (e.g. fixed points) and global (e.g. attractors) and the directionality of the dynamics organizes the data.
The topological data have associated algebraic invariants which may further codify the relationship between local and global and often recover the invariance of the dynamics, i.e. provide information about the existence and structure of the invariant sets.

§.§ Topologization and discretization

The novelty of our approach – and the first theme of this paper – is to formalize the dynamical system itself as a topology, and capture both topology and dynamics in the formalism of bi-topological spaces, i.e. a topologization of the dynamical system.
Recall that a
semi-flow on a topological space $(X,\scrT)$[A topology on $X$ is denoted by $\scrT$.]
is a continuous map $\phi\colon \R^+\times X \to X$ such that

(i) $\phi(0,x)=x$ for all $x\in X$, and

(ii) $\phi(t,\phi(s,x)) =\phi(t+s,x)$ for all $s,t\in \R^+$ and $x\in X$.

For a semi-flow $\phi$  the
backward image  is denoted by $\phi(-t,x)$, $t>0$ and is defined as $\phi(-t,x) := \{x'\in X\mid \phi(t,x) = x'\}$.

One way to regard a topological space is via a closure operator on the algebra of subsets of $X$, cf. Sect. <ref>. Using this point of view, a map $\phi$ as defined above yields a natural closure operator on $X$ through backward or forward images. If we disregard the continuity of $\phi$ in $(X,\scrT)$  this defines  Alexandrov topologies on $X$ which are denoted by $\scrT^-$ and $\scrT^+$ respectively, cf. Sect. <ref>.
The topologies $\scrT^-$ and $\scrT^+$ record directionality of the flow, but discard the sense of time and invariance. They are also independent of the continuity properties of $\phi$.
We therefore define a topology which allows one to incorporate the continuity of $\phi$ in $(X,\scrT)$ and  which is more suited for capturing the important characteristics of dynamics such as invariance. We will refer to this topology as the (derived)  denoted by $\underline\scrT^0$, cf. Sect.  <ref>.[As we have cast dynamics as topology, it is worthwhile to ask the question: what can dynamical systems theory say about topology? Section <ref> may be regarded as steps in this direction.]
The topologies $\scrT$ and $\underline\scrT^0$ comprise the bi-topological space $(X,\scrT,\underline\scrT^0)$ which becomes our model for a semi-flow $\phi$.[The topologies $\scrT$ and $\underline\scrT^0$ are in general related while the topologies $\scrT$ and $\scrT^-,\scrT^+$ are independently defined. The is not Alexandrov in general. An explanation of the notation of the various flow topologies we use is given  in Sect.'s <ref> and <ref>.]
Sets that are closed in $(X,\scrT)$ and open  $(X,\underline\scrT^0)$, so called pairwise clopen sets,    are closed attracting blocks for $\phi$, cf. Thm. <ref>.
Of course the $\underline\scrT^0$ discards some  information about $\phi$.
However, we can define suitable (co)homology theories on $(X,\scrT,\underline\scrT^0)$ which allow one to describe fundamental invariant structures of the dynamics, cf. Sect. <ref>.

The second theme of this paper concerns discretization of both topology and dynamics.
The last few decades have seen ever more sophisticated uses of discrete approximation
in order to explore global dynamical features [3, 12, 14, 35, 36, 37, 47]; these techniques are largely based on Conley theory, a topological generalization of Morse theory [13].
As before, we describe topology in terms of a closure algebra, which

provides a powerful formalism for discretization and extends these techniques.
Moreover, as we encode a semi-flow $\phi$ as a topology, topological discretization also provides a means of discretizing $\phi$.

If we describe a topological space via a closure algebra $\bigl(\sSet(X),\cl\bigr)$, then
discretization may be regarded as determining a finite sub-algebra in the category of closure algebras, cf. Sect. <ref>.
A finite closure algebra can be represented by $\bigl(\sSet(\cX),\cl\bigr)$, where $\cX$ is a finite set, and is
equivalently described by a finite pre-order $(\cX,\le)$, called the specialization pre-order
of the associated Alexandrov topology on $\cX$.
Duality of the latter pre-order defines
a continuous map

\[
\disc\colon X \twoheadrightarrow \cX,
\]

which is called the discretization map and provides a discretization of $(X,\scrT)$ by a finite topological space $(\cX,\le)$, cf. Eqn. (<ref>). The elements of $\cX$ are denoted by $\xi$ and are called cells in $\cX$. The closure algebra for a bi-topological space such as $(X,\scrT,\underline\scrT^0)$ is given by $\bigl(\sSet(X),\cl,\ucl^0\bigr)$ and by considering finite sub-structures we obtain discretization maps $\disc\colon X\to \cX$ that are continuous with respect to two finite topologies $(\cX,\le)$ and $(\cX,\le^0)$.

in 0,1,2
in 0,1,2

[black,fill opacity=.15](+,+) rectangle (+1-,+1-);

in 0,1,2,3
in 0,1,2
[black, fill opacity=1.0, thick] (,+) – (,+1-);
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in 0,1,2,3
[black, fill opacity=1.0 ,thick] (+,) – (+1-,);

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in 0,2,4,6
in 1,3,5
() at (,) [edge] ;

in 1,3,5
in 1,3,5
() at (,) [2cell] ;

[count=ξ]  in 1,3,5
[count=] in 1,3,5
[-latex] (,+) to (,+1-);
[-latex] (+,) to (+1-,);
[-latex] (-,) to (-1+,);
[-latex] (,-) to (,-1+);

[count=ξ]  in 1,3,5
[count=] in 0,2,4,6

[-latex] (+,) to (+1-,);
[-latex] (-,) to (-1+,);

[count=ξ]  in 0,2,4,6
[count=] in 1,3,5
[-latex] (,+) to (,+1-);
[-latex] (,-) to (,-1+);

[white]  (4+, -) rectangle ( 6+, 2-);

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(2,2) – (3,2);
(0,1) – (2,1);
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[Orange] (01) at (.75,1.35)[dot] ;
[Orange] (02) at (0.4,2.5)[dot] ;
[Orange] (11) at (1.25,1.65)[dot] ;
[Orange] (23) at (2,2.5)[dot] ;
[Orange] (12) at (1.025,1.5)[dot] ;

[Orange, postaction=decorate] (11) to[out=80,in=-10] (02);
[Orange, postaction=decorate] (23) to[out=180,in=50] (02);

[Orange, postaction=decorate] (01) to[out=80,in=160] (11);
[Orange, postaction=decorate] (11) to[out=270,in=340] (01);
[Orange]  (.85,1.425) to[out=90,in=120] (1.16,1.575);
[Orange] (1.16,1.575) to[out=290,in=280] (.85,1.425);

[Orange, postaction=decorate] (01) to[out=280,in=80] (00);

[Orange, postaction=decorate] (2.5,3.1) to[out=290,in=0] (23);

[Orange, postaction=decorate] (-.1,2.35) to[out=0,in=250] (02);
[Orange, postaction=decorate] (-.1,3.1) to[out=330,in=150] (02);

[Orange, postaction=decorate] (3.1,2.5) to[out=240,in=25] (02);
[Orange, postaction=decorate] (2.1,3.1) to[out=240,in=60] (02);
[Orange, postaction=decorate] (-0.1,1.65) to[out=0,in=350] (02);
[Orange, postaction=decorate] (-0.1,1.95) to[out=20,in=300] (02);

[Orange, postaction=decorate] (1.125,3.1) to[out=230,in=100] (02);

[Orange, postaction=decorate] (2.1,1.65) to[out=170,in=10] (02);

[Orange, postaction=decorate]
(3.1, 1.45) .. controls (2.75, 1.85)  and (2.55, 2.1) .. (2.2, 2.25);

[Orange, postaction=decorate] (2.4,1.75) .. controls (1.9,2.4) and (1.75,1.5)  .. (1.295, 2.18);

[Orange, postaction=decorate] (3.075,3.1) .. controls (2.5,2.25) and (2,2.25)  .. (1.75, 2.41);

[Orange, postaction=decorate] (2.025,-.1) .. controls (1.95,0.25) and (1.5,0.3)  .. (.95, .39);

[Orange, postaction=decorate] (-.1,1.5) to[out=0,in=150] (01);

[Orange, postaction=decorate] (-.1,0.5) to[out=0,in=170] (00);
[Orange, postaction=decorate] (-.15,-.05) to[out=0,in=190] (00);
[Orange, postaction=decorate] (.5,-0.1) to[out=80,in=290] (00);
[Orange, postaction=decorate] (2.1,0.4) to[out=180,in=-10] (00);

[Orange, postaction=decorate] (2.4,.75) .. controls (2,1.4) and (1.5, .6) .. (00);

[Orange, postaction=decorate] (2.1,1.35) to[out=170,in=51] (00);
[Orange, postaction=decorate] (-.1,1.35) to[out=10,in=90] (00);
[Orange, postaction=decorate] (-.1,1.03) to[out=-10,in=120] (00);
[Orange, postaction=decorate] (1.55,-.1) to[out=100,in=-30] (00);
[Orange, postaction=decorate] (1.05,-.1) to[out=110,in=-40] (00);

[Orange, postaction=decorate] (3.1,1.1) to[out=140,in=-10] (11);

[Orange, postaction=decorate] (1.795,3.1) .. controls (1.65,2.95) and (1.55,2.85)  .. (1.3, 2.825);

[Orange, postaction=decorate] (3.1,1.95) .. controls (2.65,2.15) and (2.4,2.175)  .. (2.25, 2.235);

[Orange, postaction=decorate] (2.8,.8) .. controls (2,1.2)   .. (1.35, .74);

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in 1,3,5
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in 0,2,4,6
in 1,3,5
() at (,) [edge] ;

in 1,3,5
in 1,3,5
() at (,) [2cell] ;

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[count=] in 1,5
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[-latex] (,-) to (,-1+);

[-latex] (+,+) to (+1-,+1-);
[-latex] (-,+) to (-1+,+1-);
[-latex] (-,-) to (-1+,-1+);
[-latex] (+,-) to (+1-,-1+);

in 2,3,4,6
[latex-] (-,5) to (-1+,5);

[latex-latex] (5-,5) to (5-1+,5);

[evaluate = as ξusing int(+1)] in 3,5
[latex-] (ξ-,6-) to (ξ-1+,6-1+);
[latex-] (,6-) to (,6-1+);

[evaluate = as ξusing int(+1)] in 5
[latex-] (ξ-,4+) to (ξ-1+,4+1-);
[latex-] (,4+) to (,4+1-);

[evaluate = as ξusing int(+1)] in 3
[latex-] (ξ-,0+) to (ξ-1+,0+1-);
[latex-] (,0+) to (,0+1-);

in 2,3,4
[latex-latex] (3,+) to (3,+1-);

[-latex] (3,1+) to (3,1+1-);

in 3,4
[latex-] (-, 1) to (-1+, 1);
in 1,2
[latex-latex] (+, 3) to (+1-, 3);
in 2,3
[latex-] (4-,) to (3+,);
[latex-] (4-,4-) to (3+,3+);

[latex-] (3-,3+) to (2+,4-);
[latex-] (3-,3-) to (2+,2+);

[-latex] (1,3+) to (1,4-);

[latex-] (1,3-) to (1,2+);
[latex-] (0+,3) to (1-,3);

[-latex] (4+, 2) to (5-, 2);
[-latex] (4+, 3) to (5-, 3);
[-latex] (5+, 3) to (6-, 3);
[-latex] (5, 4-) to (5, 3+);

[-latex] (4+, 4-) to (5-, 3+);
[-latex] (5+, 3-) to (6-, 2+);

[-latex] (5,3-) to (5,2+);

[white]  (4+, -) rectangle ( 6+, 2-);

A discretization of $(X,\scrT)$ with the associated face partial order $\le$ [left 1 and 2]. An example of a semi-flow $\phi$ on $X$ and pre-order $\le^0$ which is a discretization of the $\underline\scrT^0$ [right 3 and 4].

That is, for $(X,\scrT,\underline\scrT^0)$ a discretization is a finite bi-topological space $(\cX,\le,\le^0)$ with continuous discretization map $\disc\colon (X,\scrT,\underline\scrT^0)
\twoheadrightarrow (\cX,\le,\le^0)$; this procedure is described for general bi-topological spaces in Sect. <ref>.
Fig. <ref>  illustrates discretization of $(X,\scrT,\underline\scrT^0)$ via two compatible pre-orders on a finite topological  space $\cX$.

Attracting blocks, cf. Eqn. (<ref>), play a central role in the study of the gradient-like and recurrent dynamics of a semi-flow $\phi$.

As earlier noted,
the pairwise clopen sets in $(X,\scrT,\underline\scrT^0)$ comprise
the closed attracting blocks for $\phi$.

We can describe such sets in terms of a discretization which synthesizes both topologies.
A Morse pre-order, cf. Defn. <ref>, is a pre-order $(\cX,\le^\dagger)$ such that both

\[
\disc\colon (X,\scrT)\twoheadrightarrow (\cX,\le^\dagger)\quad\text{ and }\quad
\disc\colon (X,\underline\scrT^0)\twoheadrightarrow (\cX,\ge^\dagger)
\]

are continuous; Morse pre-orders  are particular instances of antagonistic pre-orders, which are defined purely in terms of bi-topological spaces, cf. Defn. <ref>.

Closed sets in $(\cX,\le^\dagger)$ correspond to closed attracting blocks for $\phi$, cf. Fig. <ref>[right].

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in 0,2,4,6
in 0,2,4,6
() at (,) [vert] ;

in 0,2,4,6
in 1,3,5
() at (,) [edge] ;

in 0,2,4,6
in 1,3,5
() at (,) [edge] ;

in 1,3,5
in 1,3,5
() at (,) [2cell] ;
[count=ξ]  in 1
[count=] in 1,5
[latex-latex] (,+) to (,+1-);
[latex-latex] (+,) to (+1-,);
[latex-latex] (-,) to (-1+,);
[latex-latex] (,-) to (,-1+);

[latex-latex] (+,+) to (+1-,+1-);
[latex-latex] (-,+) to (-1+,+1-);
[latex-latex] (-,-) to (-1+,-1+);
[latex-latex] (+,-) to (+1-,-1+);

in 3
[-latex] (-,5) to (-1+,5);

in 4,5,6
[latex-latex] (-,5) to (-1+,5);

[evaluate = as ξusing int(+1)] in 3,5
[latex-latex] (ξ-,6-) to (ξ-1+,6-1+);
[latex-latex] (,6-) to (,6-1+);

[evaluate = as ξusing int(+1)] in 5
[latex-latex] (ξ-,4+) to (ξ-1+,4+1-);
[latex-latex] (,4+) to (,4+1-);

[evaluate = as ξusing int(+1)] in 3
[latex-latex] (ξ-,0+) to (ξ-1+,0+1-);
[latex-latex] (,0+) to (,0+1-);

in 1,2,3,4
[latex-latex] (3,+) to (3,+1-);

in 3
[-latex] (-, 1) to (-1+, 1);
in 4
[latex-latex] (-, 1) to (-1+, 1);
in 1,2
[latex-latex] (+, 3) to (+1-, 3);
in 2,3
[latex-latex] (4-,) to (3+,);
[latex-latex] (4-,4-) to (3+,3+);

[-latex] (3-,3+) to (2+,4-);
[-latex] (3-,3-) to (2+,2+);

[-latex] (1,3+) to (1,4-);
[-latex] (1,3-) to (1,2+);

[latex-latex] (0+,3) to (1-,3);
[latex-latex] (5-,4) to (4+,4);

[-latex] (3-,5+) to (2+,6-);
[-latex] (3-,5-) to (2+,4+);
[-latex] (3+,5-) to (4-,4+);
[-latex] (5-,5+) to (4+,6-);
[-latex] (5-,5-) to (4+,4+);

[-latex] (1-,3+) to (0+,4-);
[-latex] (1-,3-) to (0+,2+);

[-latex] (1+,3+) to (2-,4-);
[-latex] (1+,3-) to (2-,2+);

[-latex] (3+,3-) to (4-,2+);

[latex-latex] (5+,3-) to (6-,2+);
[latex-latex] (5+,3) to (6-,3);
[latex-latex] (5,3-) to (5,2+);

[-latex] (5+,3+) to (6-,4-);
[-latex] (5,3+) to (5,4-);
[-latex] (5+,3-) to (4+,4-);
[-latex] (5-,3) to (4+,3);
[-latex] (5-,3-) to (4+,2+);
[-latex] (5-,2) to (4+,2);

[-latex] (3-,1+) to (2+,2-);
[-latex] (3-,1-) to (2+,);
[-latex] (3+,1+) to (4-,2-);

[white]  (4+, -) rectangle ( 6+, 2-);

(-,̣-)̣ – (2+,̣-)̣ – (2+,̣2+)̣ – (-,̣2+)̣ – cycle ;
(-,̣4-)̣ – (2+,̣4-)̣ – (2+,̣6+)̣ – (-,̣6+)̣ – cycle ;

(6+,̣6+)̣ – (3-,̣6+)̣ – (3-,̣3+)̣ – (0-,̣3+)̣ – (0-,̣3-)̣ – (3-,̣3-)̣ – (3-,̣-)̣ – (3-,̣-)̣ – (4+,̣-)̣ – (4+,̣ 4-)̣ – (6+,̣4-)̣ – cycle;

(5-,̣2-)̣ – (5-,̣3+)̣ – (6+,̣3+)̣ – (6+,̣2-)̣ – cycle ;

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[Orange] (01) at (.75,1.35)[dot] ;
[Orange] (02) at (0.4,2.5)[dot] ;
[Orange] (11) at (1.25,1.65)[dot] ;
[Orange] (23) at (2,2.5)[dot] ;
[Orange] (12) at (1.025,1.5)[dot] ;

at (-.25,.5) $T_0$;
at (-.25,1.5) $T_2$;
at (-.25,2.5) $T_1$;
at (3.35,1.525) $T_3$;

[Orange, postaction=decorate] (11) to[out=80,in=-10] (02);
[Orange, postaction=decorate] (23) to[out=180,in=50] (02);

[Orange, postaction=decorate] (01) to[out=80,in=160] (11);
[Orange, postaction=decorate] (11) to[out=270,in=340] (01);
[Orange]  (.85,1.425) to[out=90,in=120] (1.16,1.575);
[Orange] (1.16,1.575) to[out=290,in=280] (.85,1.425);

[Orange, postaction=decorate] (01) to[out=280,in=80] (00);

[Orange, postaction=decorate] (2.5,3.1) to[out=290,in=0] (23);

[Orange, postaction=decorate] (-.1,2.35) to[out=0,in=250] (02);
[Orange, postaction=decorate] (-.1,3.1) to[out=330,in=150] (02);

[Orange, postaction=decorate] (3.1,2.5) to[out=240,in=25] (02);
[Orange, postaction=decorate] (2.1,3.1) to[out=240,in=60] (02);
[Orange, postaction=decorate] (-0.1,1.65) to[out=0,in=350] (02);
[Orange, postaction=decorate] (-0.1,1.95) to[out=20,in=300] (02);

[Orange, postaction=decorate] (1.125,3.1) to[out=230,in=100] (02);

[Orange, postaction=decorate] (2.1,1.65) to[out=170,in=10] (02);

[Orange, postaction=decorate]
(3.1, 1.45) .. controls (2.75, 1.85)  and (2.55, 2.1) .. (2.2, 2.25);

[Orange, postaction=decorate] (2.4,1.75) .. controls (1.9,2.4) and (1.75,1.5)  .. (1.295, 2.18);

[Orange, postaction=decorate] (3.075,3.1) .. controls (2.5,2.25) and (2,2.25)  .. (1.75, 2.41);

[Orange, postaction=decorate] (2.025,-.1) .. controls (1.95,0.25) and (1.5,0.3)  .. (.95, .39);

[Orange, postaction=decorate] (-.1,1.5) to[out=0,in=150] (01);

[Orange, postaction=decorate] (-.1,0.5) to[out=0,in=170] (00);
[Orange, postaction=decorate] (-.15,-.05) to[out=0,in=190] (00);
[Orange, postaction=decorate] (.5,-0.1) to[out=80,in=290] (00);
[Orange, postaction=decorate] (2.1,0.4) to[out=180,in=-10] (00);

[Orange, postaction=decorate] (2.4,.75) .. controls (2,1.4) and (1.5, .6) .. (00);

[Orange, postaction=decorate] (2.1,1.35) to[out=170,in=51] (00);
[Orange, postaction=decorate] (-.1,1.35) to[out=10,in=90] (00);
[Orange, postaction=decorate] (-.1,1.03) to[out=-10,in=120] (00);
[Orange, postaction=decorate] (1.55,-.1) to[out=100,in=-30] (00);
[Orange, postaction=decorate] (1.05,-.1) to[out=110,in=-40] (00);

[Orange, postaction=decorate] (3.1,1.1) to[out=140,in=-10] (11);

[Orange, postaction=decorate] (1.795,3.1) .. controls (1.65,2.95) and (1.55,2.85)  .. (1.3, 2.825);

[Orange, postaction=decorate] (3.1,1.95) .. controls (2.65,2.15) and (2.4,2.175)  .. (2.25, 2.235);

[Orange, postaction=decorate] (2.8,.8) .. controls (2,1.2)   .. (1.35, .74);

The antagonistic coarsening of $\le$ and $\le^0$ in Fig. <ref> makes a Morse pre-order $\le^\dagger$ [left], with $\sSC$ depicted as the outlined sets. The realization yields a Morse tessellation $(\sT,\le)$ [right].

The lattice $\sO(\cX,\le^\dagger)$ of closed sets in $(\cX,\le^\dagger)$  can be represented by the down-sets of a finite poset $(\sSC,\le)$, cf. Eqn. (<ref>), Fig. <ref>[left]. The map $\dyn\colon\! \cX\! \twoheadrightarrow\! (\sSC,\le\!)$ is defined as the dual of $\sO(\cX,\le^\dagger) \cong \sO(\sSC,\le) \rightarrowtail \sSet(\cX)$, cf. Eqn.'s (<ref>)-(<ref>).[This map is christened $\dyn$ as it may be regarded as a grading by the dynamics.  Combining (<ref>) and (<ref>)

provides a formula for $\dyn$, q.v. Thm. <ref> and Eqn. (<ref>).]
Depending on the topology on $\cX$ the map $\dyn$ is order-preserving, or order-reversing[The dashed arrows in Figure <ref> indicated order-reversing maps. We use this notation throughout the paper.] as is displayed in the following diagram:

\begin{equation}
\dgARROWLENGTH=2em
\begin{diagram}
\node{}\node{(\cX,\le)}\arrow{se,l}{}\arrow{seee,l}{\dyn}\node{}\node{}\\
\node{X}\arrow{ne,l}{\grd}\arrow{se,r}{\grd}\node{}\node{(\cX,\le^\dagger)}\arrow[2]{e,l}{\dyn}\node{}\node{\sSC~~}\\
\node{}\node{(\cX,\le^0)}\arrow{ne,l,--}{}\arrow{neee,r,--}{\dyn}\node{}\node{}
\end{diagram}\label{dia:intro}
\end{equation}

The composition $X\xrightarrow{\disc} \cX\xrightarrow{\dyn} \sSC$ yields the $T_0$-discretization
$\tile\colon X\to \sSC$, cf. App. <ref>;

in particular,
$\tile\colon (X,\scrT) \to (\sSC,\le)$ and
$\tile\colon (X,\underline\scrT^0)\to (\sSC,\ge)$ are continuous.

The discretization $\tile$
defines a Morse tessellation with locally closed tiles $T=\tile^{-1}(\cS)$, $\cS\in \sSC$,
i.e. the sets $T$ form a tessellation of $X$ such that $\big\downarrow T$ is closed and $\phi(t,x)\in \Int \big\downarrow T$ for every $x\in T$ and for all tiles $T$,
cf. Defn. <ref> and <cit.>.
Fig. <ref>[right] illustrates how a Morse tessellation is obtained from a Morse pre-order.
Conversely, if $(\sT,\le)$ is a Morse tessellation we obtain a Morse pre-order by defining $(\sT,\le)$ to be a Morse pre-order, and thus a discretization of $(X,\scrT,\underline\scrT^0)$, cf. Sect. <ref>.

For Boolean discretizations, i.e. discretizations for which the closure operators for $(X,\scrT)$ and $(\cX,\le)$ commute,[For example, CW-decompositions.] cf. Defn. <ref>, we can exploit the fact that sets in $\sO(\cX,\le^\dagger)$ correspond to regular closed sets, cf. Thm. <ref>. In this case we can define a pre-order $\le^\topc$ on the top cells[Top cells are elements in $(\cX,\le)$ that are maximal with respect to $\le$.] $\xi^\topc\in \cX^\topc$ such that $\sO(\cX^\topc,\le^\topc) \cong \sO(\cX,\le^\dagger)$.
Such a pre-order $(\cX^\topc,\le^\topc)$ is called a discrete resolution  and drastically reduces the amount of data to analyze, cf. Sect.'s <ref> and <ref>.[In the application to parabolic systems in Section <ref> one may  achieve a data reduction of orders of magnitude using discrete resolutions, e.g. for a cubical CW-decomposition of a $d$-cube we have $4^{-d} < |\cX^\topc|/|\cX| < 2^{-d}$.

Discrete resolutions do not lose information about the initial Morse pre-order and associated Morse tessellation. The map $\cU^\topc \mapsto \cl~\cU^\topc$, with $\cU^\topc\in \sO(\cX^\topc,\le^\topc)$, defines an injective lattice homomorphism $\cl\colon\sO(\cX^\topc,\le^\topc) \rightarrowtail \sO(\cX,\le)$, whose image

$\sO(\cX,\le^\dagger)$ is the lattice of down-sets of the Morse pre-order, and yields the factorization:

\[
\begin{diagram}
\node{\sO(\cX^\topc,\le^\topc)}\arrow{e,l,<>}{\cong}\arrow{se,r,<>}{\cong}\node{\sO(\cX,\le^\dagger)}\arrow{s,r,<>}{\cong}\arrow{e,l,V}{}\node{\sO(\cX,\le)}\arrow{e,l,V}{}\node{\sSet(\cX)}\\
\node{}\node{\sO(\sSC,\le)}\arrow{ne,l,V}{}\arrow{nee,r,V}{\dyn^{-1}}
\end{diagram}
\]

cf. Thm. <ref>.
The discretization $\dyn\colon (\cX,\le) \to(\sSC,\le)$ is the map that allows us to alternate between Morse pre-orders and discrete resolutions, cf. Sect. <ref>, and is dual to the injection $\cl\colon\sO(\cX^\topc,\le^\topc) \rightarrowtail \sO(\cX,\le)$.  A formula for $\dyn$ is given in Theorem <ref>.
In summary, $(\cX,\le^\dagger)$ is the relevant topology that contains the information about closed attracting blocks. A discrete resolution $(\cX^\topc,\le^\topc)$ is a coarser data structure  than  $(\cX,\le^\dagger)$ and which yields the same down-sets, cf. Fig. <ref>.
The Morse pre-order can be retrieved from the discrete resolution, cf. Thm. <ref>. In Sections <ref> and <ref> we exploit this principle in the application to parabolic systems.

[rectangle, draw=black,fill opacity=.2, text opacity=1] (00) at (0, 0)$\xi_0 $;
[rectangle, draw=black,fill opacity=.2, text opacity=1] (01) at (0, 1)$ \xi_2 $;
[rectangle, draw=black,fill opacity=.2, text opacity=1] (02) at (0, 2) $\xi_1 $;

[rectangle, draw=black,fill opacity=.2, text opacity=1] (10) at (1, 0)$\xi_3 $;
[rectangle, draw=black,fill opacity=.2, text opacity=1] (11) at (1, 1)$\xi_4 $;
[rectangle, draw=black,fill opacity=.2, text opacity=1] (12) at (1, 2)$\xi_5 $;

[rectangle, draw=black,fill opacity=.2, text opacity=1] (22) at (2, 2) $\xi_6 $;
[rectangle, draw=black,fill opacity=.2, text opacity=1] (21) at (2, 1) $\xi_7 $;

[-latex,thick] (12) to (02);
[-latex ,thick] (10) to (00);
[latex-latex ,thick] (01) to (11);
[latex-latex ,thick] (10) to (11);
[latex-latex ,thick] (11) to (12);
[latex-latex ,thick] (12) to (22);
[-latex,thick] (01) to (02);
[-latex,thick] (01) to (00);

[-latex,thick] (21) to (11);
[-latex,thick] (21) to (22);

[node distance=.25cm and .25cm]

[draw, ellipse] (0) at (, 0) $\cS_1: \{\xi_1\}$;
[draw, ellipse] (1) at (,̱ 0) $\cS_0: \{\xi_0\}$;
[draw, ellipse] (2) at (,̧1) $\cS_2: \{\xi_i\}_{2\leq i \leq 6}$;
[draw, ellipse] (3) at (,̧2) $\cS_3: \{\xi_7\}$;

[->,>=stealth,thick] (2) to (0);
[->,>=stealth,thick] (2) to (1);
[->,>=stealth,thick] (3) to (2);

An associated discrete resolution $\le^\topc$ on the top-cells $\cX^\topc$ of CW-decomposition in Fig. <ref> [left] and the poset $\sSC$ of its partial equivalence classes [right], which is order-isomorphic to the poset  $(\sT,\le)$.

§.§ Algebraization

The above constructions have reduced dynamics to a (continuous) discretization $\tile\colon X\to \sSC$ of the space $X$. A discretization as such captures robust directionality properties of a flow. However, information about invariance is lost.
The third theme in this paper is the algebraization of semi-flows; that is, the order-theoretic structures that encode the directionality are to be augmented with (co)homological data which carry information about (robust) invariant dynamics via Wazewski's principle.

A discretization $\tile\colon X\to \sSC$, or equivalently a Morse tessellation, gives rise to a filtering of $X$ consisting of regular closed
attracting blocks, i.e. a lattice homomorphism $\alpha \mapsto F_\alpha X$, where $\alpha\in \sO(\sSC)$ is a down-set in $\sSC$ and $F_\alpha X = \tile^{-1}(\alpha)\in \sABlockR(\phi)$ is an attracting block.  In this case, connection matrix theory [20, 21, 22, 29, 52] describes a

$\sSC$-graded chain complex $\bigl(C^\tile(X),\cm^\tile \bigr)$ whose grading is given by

$C^\tile(X)=\bigoplus_{\cS\in \sSC} H(F_{\downarrow\cS} X/F_{\downarrow\cS^\pred} X)$,  where $H$ is the singular homology functor, cf. Fig. <ref>[left], cf. App. <ref>.
The differential $\cm^\tile$
is called a connection matrix for  $\tile$ and from the chain complex
$\bigl(C^\tile(X),\cm^\tile \bigr)$ all homologies $H(F_\beta X/F_\alpha X)$, with $\alpha,\beta\in\sO(\sSC)$, can be computed as homology of the sub-quotient chain complex $G_{\beta\smin\alpha} C^\tile(X)$ which is denoted by $H^\tile(\beta\smin\alpha)$.  This data can be visualized as a poset isomorphic to $\sSC$,
whose elements are pairs $\bigl(\cS,P_\mu(\cS)\bigr) \in \sSC\times \Z_+[\mu]$, where $P_\mu(\cS)$
is the Poincaré polynomial of $H^\tile(\cS)$ which uses the natural dimension grading of singular homology,

cf. Fig. <ref>[right].

Such a poset  will be referred to as the tessellar phase diagram[The tessellar phase diagram is expressed with respect to Borel-Moore homology.]
$(\tessph,\le^\dagger)$ for  $(\cX,\le^\dagger)$, cf. Sect. <ref>.

[column sep = 1.5em]
0 r
_2⟨_2⟩r_1 =

[node distance=.2cm and .25cm]

[draw, ellipse] (0) at (, 0) $\cS_1 : \mu^0$;
[draw, ellipse] (1) at (,̱ 0) $\cS_0 : \mu^0$;
[draw, ellipse] (2) at (,̧1) $\cS_2 : \mu^1$;
[draw, ellipse] (3) at (,̧2) $\cS_3 : 0$;

[->,>=stealth,thick] (2) to (0);
[->,>=stealth,thick] (2) to (1);
[->,>=stealth,thick] (3) to (2);

The $\sSC$-graded  chain complex  $(C^\tile(X),\cm^\tile)$  [left] of Fig. <ref> (computed using $\Z_2$ coefficients) and associated the tessellar phase diagram $(\tessph,\le^\dagger)$ [right]. We display the pair $(\cS,P_\mu(\cS))$ as $\cS$ and $P_\mu(\cS)$ as a matter of style.

A discretization $\tile\colon X\to \sSC$ may be considered purely from a topological perspective, independent of dynamics.  Given a sufficiently `nice' discretization, the connection matrix theory can be applied to $\tile$.   We regard this as part of the synthesis of dynamics and topology, and using dynamical tools to analyze topology.  In Section <ref>, we show how this leads to a homology theory which we call tessellar homology and which, in contradistinction to cellular homology, uses general tiles instead of CW-cells. In Section <ref> we show how cellular homology specifically is recovered.

One of the key advantages of using Morse pre-orders is that $\tile$ factors through the discretization of $(X,\scrT)$, which enables a computational approach to connection matrix theory using the algorithms of [29, 30] and associated software [31].
For the sake of the simplicity, we explain this in the case that $(X,\scrT)$ admits a finite CW-decomposition.

The idea is encapsulated in the following diagram, cf. Sect. <ref>:

\begin{equation}
\label{factor3ind}
% \dgARROWLENGTH=1.8em
% \begin{diagram}
%     \node{} \node{(\cX,\leq)}\arrow{s,r}{\dyn}\arrow{e,l}{\pi}\node{\cX/_\sim} \arrow{sw,r}{\varpi}\\
%     \node{X}\arrow{e,l}{\tile}\arrow{ne,l}{\cell} \node{\sSC}
% \end{diagram}
% \quad\quad
\dgARROWLENGTH=3.5em
\begin{diagram}
\node{} \node{(\cX,\leq)}\arrow{s,r}{\dyn}\\%\arrow{se,l}{}\\
%\node{\cX/_\sim} \arrow{sw,r}{}\\
\node{X}\arrow{e,l}{\tile}\arrow{ne,l}{\cell} \node{\sSC}\arrow{e,l}{\ind} \node{\Z}
\end{diagram}
\end{equation}

In the case of a CW-decomposition map $\cell$, the associated pre-order is a partial order (face partial order) and the above diagram, cf. (<ref>), provides the factorization via $(\cX,\le)$.

The fact that we factor $\tile$ through $(\cX,\le)$ allows us to compute $(C^\tile(X),\cm^\tile)$ by instead computing connection matrices in two  simpler settings. First for the discretization $\cell\colon X\to (\cX,\le)$ to obtain the cellular chain complex $C^\cell(X)$ graded over the poset $\cX$ of cells which represents the singular homology as cellular homology; then a second time by re-grading $C^\cell(X)$ by $\sSC$ via $\dyn\colon \cX\to \sSC$ and using the algorithm of [29] to produce $(C^\tile(X), \cm^\tile)$.

As indicated above the discretization $\tile\colon X\to\sSC$ is  equivalent to a Morse tessellation $(\sT,\le)\xrightarrow[]{\cong} (\sSC,\le)$, where the tiles $T\in \sT$ are given by $T=\tile^{-1}(\cS)$.
Since $\{\cS\} = \beta\smin\alpha$, the homology of $T$ is given by

\[
H^\tile(\beta\smin\alpha) \cong H^\cell(G_{\beta\smin\alpha}\cX) \cong H^\BM(T),
\]

cf. Thm. <ref>, where the latter is the Borel-Moore homology of $T$, cf. [10, 11, 24, 28, 27, 33]. Since $T$ is the set-difference of two attracting blocks for $\phi$ it is an isolating block (neighborhood)
for $\phi$, cf. [40], and $H^{\tile}(\beta\smin\alpha)$ represents  the Conley index of $T$.

An additional scalar discretization $\ind\colon\sSC\to \Z$, cf. (<ref>), allows a second grading of $H^\tile$ which make $H^\tile$ a bi-graded homology theory denoted by $H^\tile_{p,q}(\cX)$, cf. Sect. <ref>.
The discretization $\ind$ induces a spectral sequence which allows an additional $\Z$-grading of the tessellar homology in accordance with the $\sSC$-grading of $C^\tile$. As a matter of fact $H^\tile_{p,q}(\cU\smin\cU')$ is well-defined for any convex set $\cU\smin\cU'$ in $\sSC$.
In this context $(C^\tile,\cm^\tile)$ may be regarded as chain complex as well as $\Z$-graded differential module. If we define the Poincaré polynomials
P_{\lambda,\mu}(C^\tile) = \sum_{\cS\in \sSC}P^\tile_{\lambda,\mu}(\cS)
$ with

$P^\tile_{\lambda,\mu}(\cS) = \sum_{p,q\in \Z} \bigl(\rk H^\tile_{p,q}(\cS)\bigr) \lambda^p \mu^q$
and $P_{\lambda,\mu}^\tile(\cX) = \sum_{p,q\in \Z} \bigl(\rk H^\tile_{p,q}(\cX)\bigr) \lambda^p \mu^q$. Then the following
variation on the standard Morse relations is satisfied, cf. Thm. <ref>,

\begin{equation*}
\sum_{\cS\in \sSC}P^\tile_{\lambda,\mu}(\cS)
= P_{\lambda,\mu}^\tile(\cX) + \sum_{r=1}^\infty (1+\lambda^r \mu) Q^r_{\lambda,\mu},
\end{equation*}

where $Q^r_{\lambda,\mu} \ge 0$ and the sum over $r$ is finite.
The $p$-index is a manifestation of the and the $q$-index of the phase space topology making it a true tale of two topologies. The impact is most apparent in the application to parabolic flows where we use a canonical discretization $\lap \colon \sSC \to \Z$.

§.§ Parabolic flows and braid invariants

Sections <ref>–<ref> encompass  the final theme  wherein we demonstrate the ideas and methods in this paper for a large class of flows,
called discrete parabolic flows.

Such flows occur in a wide variety of settings, e.g. studying the infinite dimensional dynamics of scalar parabolic equations which can be realized via discrete parabolic equations, cf. [25], [5].
Discrete parabolic equations and parabolic flows

also play a prominent role in the theory of monotone twist maps whose dynamics can be
studied using parabolic flows, cf. [2], [15].
The introduction of Morse theory on braids in [26] was sparked by questions for fourth order Lagrangian dynamics which use parabolic flows

to describe periodic solutions. Finally,  parabolic flows play a pivotal role
in computing braid Floer homology, cf. [6].

A parabolic flow $\phi$ is defined via

\dot x_i = R_i(x_{i-1},x_i,x_{i+1}),
with $R_{i+d}=R_i$,
where the (smooth) functions $R_i(x_{i-1},x_i,x_{i+1})$ are monotone with respect to their first and third argument and their stationary equations $R_i(x_{i-1},x_i,x_{i+1})=0$ are referred to as parabolic recurrence relations.
Parabolic recurrence relations form a perfect symbiosis
with discretized braids.
A discretized braid (diagram) $x$ on $n$ strands and $d$ discretization points is a unordered collection of sequences $\cdots, x_0^\alpha,x_1^\alpha,\cdots$ such that
$x_{i+d}^{\theta(\alpha)} = x_i^\alpha$, for a permutation $\theta\in S_n$, and $\alpha=1,\cdots,n$.

[dot/.style=draw,circle,fill,inner sep=.75pt,line width=.7pt,scale=.65]

in 0,1,2,3,4,5,6
in ,,̱,̧,̣
() at (, )[dot] ;

in ,
in 0,1,2,3,4,5
(,) to (+1,);

in 0,2,3,5
(, )̱ to (+1, )̧;
(,)̧ to (+1,)̣;
(,)̣ to (+1,)̱;

in 1, 4
(,)̣ to (+1,)̧;
(,)̧ to (+1,)̱;
(,)̱ to (+1,)̣;

[scale=.65,node distance=.3cm and .25cm]

[draw, ellipse] (0) at (, 0) $\cS_0 : \lambda^0 \mu^0$;
[draw, ellipse] (1) at (,̧ 0) $\cS_1 : \lambda^0 \mu^0$;

[draw, ellipse] (4) at (, 3.25*) $\cS_4 : \lambda^5 \mu^3$;
[draw, ellipse] (5) at (,̧ 3.25*) $\cS_5 : \lambda^5 \mu^3$;

[draw, ellipse] (7) at (, 5.2*) $\cS_7 : \lambda^7 \mu^5$;
[draw, ellipse] (8) at (,̧ 5.2*) $\cS_8 : \lambda^7 \mu^5$;

[draw, ellipse] (2) at (,̱ ) $\cS_2 : \lambda^3 \mu^1+ \lambda^3 \mu^2$;
[draw, ellipse] (3) at (,̱ 2.3*) $\cS_3 : \lambda^4 \mu^2$;
[draw, ellipse] (6) at (,̱ 4.25*) $\cS_6b : \lambda^6 \mu^4$;
[draw, ellipse] (9) at (,̱ 6.2*) $\cS_9 : \lambda^8 \mu^6$;

[->,>=stealth,thick] (2) to (0);
[->,>=stealth,thick] (2) to (1);
[->,>=stealth,thick] (3) to (2);
[->,>=stealth,thick] (4) to (3);
[->,>=stealth,thick] (5) to (3);
[->,>=stealth,thick] (6) to (4);
[->,>=stealth,thick] (6) to (5);
[->,>=stealth,thick] (7) to (6);
[->,>=stealth,thick] (8) to (6);
[->,>=stealth,thick] (9) to (7);
[->,>=stealth,thick] (9) to (8);

A discretized (pseudo-anosov) braid $y$ [left], and associated reduced tessellar phase diagram $\otessph(y)$ [right]. The vertices in   $\otessph(y)$ contain the Poincaré polynomials of the bi-graded parabolic homology, cf. Sect. <ref>.

By viewing such sequences by piece wise linear interpolations the various strands `intersect', cf. Fig. <ref>[left]. There is a non-degeneracy condition on the intersections (no tangencies).
A collection of  stationary solutions of $\phi$ of integer period  forms a discretized braid diagrams denoted by $y$ and referred to as a skeleton, cf. Fig. <ref>[left].

If we add an additional
periodic sequence $x$  the ensemble of $x$ and $y$  generically forms a braid $x\rel y$, called a relative discretized braid. Since $y$ is stationary for $\phi$  we may consider $\phi(t,x)\rel y$. Whenever $\phi(t,x)\rel y$ becomes singular, i.e. strands develop tangencies,  then the total number of intersections of $x$ with the strands in $y$ decreases strictly.
This principle is crucial and emphasizes the intimate relation between parabolic flows and discretized braids.
As such relative braids can be partially ordered using parabolic flows which form the backbone of the canonical discretization of the for parabolic flows.
We  construct special CW-decompositions, cf. Sect. <ref>, and discretizations of the via
appropriately defined discrete Lyapunov functions,
which allows us to establish Morse tessellations and Morse representations, cf. Sect. <ref>.

In the language of discrete resolutions the partial order on the relative braid classes defines the poset $(\sSC,\le)$ which comes from the canonical CW-decomposition of $X$ given by the skeleton $y$ and the given by $\phi$.

In   Section <ref> we give an overview of the algorithmic steps in computing the order structures and connection matrices and in Section <ref> we discuss the bi-graded parabolic homology.
In Section <ref>

we show that the parabolic differential module (and the associated  tessellar phase diagrams) we obtain for parabolic flows extend the results in [26] and provide an  invariant for positive braids which also defines a new invariant for scalar parabolic equations, cf. Thm. <ref>.

every label/.append style = font =
[column sep=2.5em, row sep =3.5em]
0 r

⊕_i= 7,8_2⟨_i⟩r_5 =
1    1

[phantom, near start,""coordinate, name=Z]d     [1.5em]
rounded corners,
to path=
– ([xshift=2ex].east)
|- (Z) [near start]-| ([xshift=-2ex].west)
– ()

⊕_i=4,5_2⟨_i⟩r_3 =
0   1

1   1

⊕_i=2,3 _2⟨_i⟩r_2 =
0    0

⊕_i=0,1 _2⟨_i⟩r_0=0     0

Associated  complex $\scrA(\beta)$, computed over $\K=\Z_2$ coefficients, viewed as chain complex using dimension grading. The index $q$
for $\cm_q$ reflects the natural grading induced by singular homology.

The importance of Theorem <ref> is that we obtain complete insight in the homology of loop spaces for scalar parabolic equations as well as the boundary homomorphisms which contain information about connecting orbits for parabolic equations.
To best explain the discretization of topology and dynamics and the main statement of Theorem <ref> we consider an example. Let $y$ be a discretized braid $y$ given in Figure <ref>[left]
and $\phi$ is any parabolic flow for which $y$ is stationary. This can be rephrased in terms of a bi-topological space for which $y$ defines a natural discretization. The  sub-poset of  homologically non-trivial braid classes in $(\sSC,\le)$ is given by the reduced tessellar phase diagram $(\otessph,\le^\dagger)$
displayed in Fig. <ref>[right].

every label/.append style = font =
[column sep=2.5em, row sep =3.5em]
0 r

⊕_i= 7,8_2⟨_i⟩r^1_7 =
1    1

[phantom, near start,""coordinate, name=Z]d     [1.5em]
rounded corners,
to path=
– ([xshift=2ex].east)
|- (Z) [near start]-| ([xshift=-2ex].west)
– ()

⊕_i=4,5_2⟨_i⟩r^1_5 =
1   1

[to path= – ([yshift=-3.5ex].south) -| (),
rounded corners]rr

_2⟨_3⟩r^1_4 =

[phantom]d[pos=0.35] ^2_5 = 0   0
0   1

1   0

1   1

⊕_i=0,1 _2⟨_i⟩r     0

The differential module $\scrA(\beta)$, computed over $\K=\Z_2$ coefficients,
with lap number grading.

For the entry $\cm_p^r$ the index $p$ reflects the lap number and $r$ indicates the distance between lap numbers.

The more detailed information is given by the parabolic differential module $\scrA(\beta)$ generated by $\tile\colon X\to \sSC$ and the scalar discretization $\lap\colon \sSC\to \Z$, which counts the intersections of $x$ with $y$ divided by two, cf. Sect. <ref> for a detailed definition.
As described in Section <ref>
$\lap$ provides a scalar grading on the tessellar homology $H^\tile(\cU\smin\cU')$, with $\cU\smin\cU'$ convex in $\sSC$, which yields the parabolic homology $\tH_{p,q}(\cU\smin\cU')$ where

\[
H^\tile(\cU\smin\cU') = \bigoplus_{p,q\in\Z} \tH_{p,q}(\cU\smin \cU').
\]

Figure <ref>[right] lists the Poincaré polynomials of the non-trivial tiles with respect to parabolic homology.
The differential module $\scrA(\beta)$
may be  regarded as a chain complex in Figure <ref> with homology
$H^\BM_q(\cS) =\oplus_{p\in \Z}\tH_{p,q}(\cS)$,
or as $\Z$-graded differential module in Figure <ref> with homology
$\tH_{p,*}(\cS) =\oplus_{q\in \Z}\tH_{p,q}(\cS)$.
By making further specification of the entries in $\cm^\tile$ the differential module can also be represented as to show the $\otessph$-order, cf. Sect. <ref>.

Theorem <ref> shows that the parabolic differential module, and thus the reduced tessellar phase diagram is a topological invariant for the topological braid $\beta(y)$, i.e. all braid diagrams isotopic to $y$, cf. Sect. <ref>. Due to Theorem <ref> the parabolic differential module and reduced tessellar phase diagram can be denoted by  $\scrA(\beta)$ and $\otessph(\beta)$  respectively.

In summary, the application of the methods put forth in this paper to parabolic flows gives a novel approach towards computing algebraic topological information about infinite dimensional problems.

The themes throughout this paper touch upon many topics. Section <ref> concludes with a discussion of related remarks and open problems.

§ DISCRETIZATION OF TOPOLOGY

Our philosophy is that discretization is the study of appropriate finite substructures.
In this section we start with an exposition of topological spaces and discretization in terms of Boolean algebras with operators, cf. App. <ref>.

§.§ Closure algebras

Let $(X,\scrT)$ be a topological space and let $\sSet(X)$ denote the Boolean algebra of subsets of $X$.   For $(X,\scrT)$ we define an associated  closure algebra  as the pair $\bigl(\sSet(X),\cl \bigr)$, where $\cl\colon \sSet(X)\to \sSet(X)$, is defined as the closure of a subset, cf. [46].
In general, an operator $\cl\colon\sSet(X) \to \sSet(X)$ is a closure operator if all four Kuratowski axioms for a closure operator are satisfied: for all $U,U'\subset X$,

(K1) (normal) $\cl~\varnothing = \varnothing$;

(K2) (additive) $\cl(U\cup U') = \cl~U \cup \cl~U'$;

(K3) (sub-idempotent) $\cl\bigl(\cl~ U\bigr) \subset \cl~ U$;[Axiom (K3) in combination with Axiom (K4) this implies that
$\cl$ is idempotent, i.e. $\cl\bigl(\cl~ U\bigr) = \cl~U$.]

(K4) (expansive) $U \subset \cl~ U$.[The single condition (K): $U\cup \cl~U\cup \cl\bigl(\cl~ U'\bigr) = \cl(U\cup U')\smin \cl~\varnothing$ is equivalent to (K1)-(K4).]

A discretization of topology in the spirit of closure algebras is
an injective  Boolean homomorphism[ In the context of discretizaztion we consider injective homomorphisms with finite range, i.e. finite subalgebras. The theory can be phrased in more general terms via homomorphisms.]   $|\cdot|\colon \sSet(\cX)\rightarrowtail \sSet(X)$, where $\sSet(\cX)$ is the
powerset of a finite set $\cX$, in combination with an appropriately chosen closure operator $\cl\colon \sSet(\cX)\to \sSet(\cX)$ such that $\cl|\cU|\subset |\cl~ \cU|$ for all $\cU\in \sSet(\cX)$. In this case we refer to
$|\cdot|\colon \sSet(\cX)\to \sSet(X)$ as a semi-homomorphism of closure algebras, cf. [7].
We refer to the elements $\xi\in \cX$ as cells.

This discretization is captured by the following (not necessarily commutative[If this diagram commutes then $|\cdot|$ is an injective homomorphism of closure algebras.]) diagram in the category of closure algebras and semi-homomorphisms:[In the above mentioned  category of closure algebras we employ the morphisms are semi-homomorphisms of closure algebras.]

\begin{equation}\label{dia:disctop}
\begin{diagram}
\node{\sSet(X)} \arrow{e,l}{\cl}\node{\sSet(X)}\\
\node{\sSet(\cX)}\arrow{n,l,V}{|\cdot|} \arrow{e,l}{\cl}\node{\sSet(\cX)}\arrow{n,r,V}{|\cdot|}
\end{diagram}
\end{equation}

The closure algebra $(\sSet(\cX),\cl)$ defines a finite topology by declaring $\cU$ closed if and only if $\cl~\cU =\cU$.  As $\cX$ is finite any topology on $\cX$ is necessarily an Alexandrov topology,
and is equivalent to a pre-order $(\cX,\le)$, called the specialization pre-order,[The Alexandrov topology is $T_0$ if and only if the specialization pre-order is
a partial order.]  defined by

\begin{equation}
\label{spec1}
\xi\le \xi' \quad\text{if and only if}\quad  \xi\in \cl\{\xi'\},
\end{equation}

cf. App. <ref>.

The discretization described above allows us to regard $\cX$ as an algebra $\bigl(\sSet(\cX),\cl\bigr)$, as a pre-order $(\cX,\le)$, and as topological space $\cX$.

In general we do not differentiate between the specialization pre-order and the Alexandrov topology, and we refer to the triple $(\cX,\cl,|\cdot|)$ as a closure algebra discretization, or CA-discretization of $X$.

A CA-discretization is called Boolean if $\cl|\cU| = |\cl~\cU|$ for all $U\in \sSet(\cX)$.

This definition in particular implies that (<ref>) is commutative in which case the  map $|\cdot|\colon \sSet(\cX)\to \sSet(X)$ is called a homomorphism of closure algebras, cf. [7].

§.§ Derivative operators and derivative algebras

Closely related to a closure operator is the notion of a derivative operator. An operator $\der\colon \sSet(X) \to \sSet(X)$ is called a derivative operator if following axioms are satisfied: for all $U,U' \subset X$,

(D1) (normal) $\der \varnothing= \varnothing$;

(D2) (additive) $\der (U\cup U') = \der U \cup \der U'$;

(D3) (quasi-idempotent) $\der(\der U)\subset U\cup \der U$.[Axiom (D3) is equivalent to from Axiom (K4) via $\cl=\id\cup\der$.]

The pair $(\sSet(X),\der)$ is called a derivative algebra, cf. [18], [46].
A derivative operator defines a closure operator via

\begin{equation}
\label{clfromder}
\cl~U:= U\cup \der U.
\end{equation}

For every closure operator there exists a derivative operator such that (<ref>) holds, e.g. take $\der = \cl$.[The choice of a derivative operator is clearly not unique. An important non-trivial choice is given by the derived set, the set of limit points of a set $U$:

\[
\der U := \bigl\{ x\in X\mid N\cap U\smin\{x\}\neq \varnothing\text{~for all neighborhoods~} N\ni x\bigr\},
\]

which is not equal to $\cl~U$ in general.
Discretization in terms of closure algebras  can always be formulated in terms of derivative algebras. A triple $(\cX,\der,|\cdot|)$, with $\der\colon\sSet(\cX) \to \sSet(\cX)$ a derivative operator, is a derivative algebra discretization, or DA-discretization of $X$ if $\der |\cU| \subset |\der \cU|$ for all $\cU\in \sSet(\cX)$ which is expressed in the diagram:

\begin{equation}\label{dia:disctop2}
\begin{diagram}
\node{\sSet(X)} \arrow{e,l}{\der}\node{\sSet(X)}\\
\node{\sSet(\cX)}\arrow{n,l,V}{|\cdot|} \arrow{e,l}{\der}\node{\sSet(\cX)}\arrow{n,r,V}{|\cdot|}
\end{diagram}
\end{equation}

This induces an
associated closure
operator $\cl\colon \sSet(\cX) \to \sSet(\cX)$
defined by $\cl = \id \cup \der$ and which satsifies
$\cl |\cU| \subset |\cl~ \cU|$, for all $\cU\in \sSet(\cX)$. This shows that
$(\cX,\cl,|\cdot|)$ is a CA-discretization.

The advantage of using a non-trivial derivative operator is a refinement of the specialization pre-order on $\cX$ in terms of an induced binary relation:

\begin{equation}
\label{spec111}
\xi < \xi' \quad\text{if and only if}\quad  \xi\in \der\{\xi'\},
\end{equation}

which is called a reduced specialization pre-order.
Observe that $<$ need not be transitive and is not reflexive in general. Moreover, the reflexive, transitive closure of $<$ is the specialization pre-order $\le$ in (<ref>). Binary relations coming from derivative operators will be called weakly transitive, or wK4, cf. [18]. If $\der$ satisfies the stronger sub-idempotency axiom in  (K3), i.e.  $\der(\der U)\subset \der U$,
we say that $\der$ is a strong
derivative operator. This is often the case in dynamics in which instance the associated reduced specialization pre-order is transitive (a K4-order).

§ DISCRETIZATION MAPS

Let $(X,\scrT)$ be a topological space.
A discretization map on $X$ is a surjective map[ As pointed in Footnote <ref>, in the context of discretization the maps are chosen to be surjective and are dual to injective closure algebra homomorphisms.]

\begin{equation}
\label{spec0}
\grd\colon X \xtwoheadrightarrow{~} \cX,
\end{equation}

where $\cX$ is a finite set. Since unions and intersections are preserved under preimage, $\grd^{-1}\colon \sSet(\cX)\to \sSet(X)$, is an injective Boolean homomorphism, cf. Footn.'s <ref> and <ref>.  In this context, we say that $\grd^{-1}$ is an evaluation map and we use the notation: $|\cU|:=\grd^{-1}(\cU)$ for $\cU\in \sSet(\cX)$.

§.§ Topology consistent pre-orders

As pointed out above any topology on $\cX$ is equivalent to its specialization pre-order $(\cX,\leq)$.  We say that $\leq$ is a $\scrT$-consistent pre-order on $X$ with respect to $\grd$ if $\grd\colon X\twoheadrightarrow (\cX,\leq)$ is continuous. The continuity of $\grd$ is equivalent to the condition that
$\cl~ \grd^{-1}(\cU) \subset \grd^{-1}(\cl~ \cU)$, for all $\cU\subset \cX$, i.e. $\cl|\cU|\subset |\cl~\cU|$.  Thus, when $\leq$ is $\scrT$-consistent, the triple $(\cX,\cl,|\cdot|)$ is a CA-discretization, and $\disc\colon (X,\scrT) \twoheadrightarrow (\cX,\le)$ is a continuous discretization map.
If $\grd$ is a continuous open map then  $|\cdot|=\grd^{-1}$ is a homomorphism of closure algebras.[By (<ref>) $\xi\le \xi'$ if and only if $\xi\in \cl~\xi'$ which is equivalent to $\{\xi\}\subset \cl~ \xi'$. In the case that $|\cdot|$ is an injective homomorphism of closure algebras we obtain the equivalent statement $|\xi|\subset |\cl~\xi'| = \cl|\xi'|$. Here we use the convention $\cl\{\xi\} = \cl~\xi$.]
For a reduced specialization pre-order $(\cX,<)$ the relation $<$ is $\scrT$-consistent if
the reflexive, transitive closure $\le$ is $\scrT$-consistent.

Conversely, given a CA-discretization $(\cX,\cl,|\cdot|)$, defining $\grd\colon X\twoheadrightarrow \cX$ via

\begin{equation}
\label{induceddisc}
\grd(x) = \xi \text{ where } x\in |\xi|,
\end{equation}

is a well-defined map since $\bigcup_\xi |\xi|=X$ and $|\xi|$ and $|\xi'|$ are mutually disjoint for all $\xi\neq\xi'$. By Birkhoff duality the injectivity of $|\cdot|$ implies the surjectivity of $\disc$, cf. Thm. <ref> and <cit.>.  Moreover, since $\cl|\cU| \subset |\cl~ \cU|$, $|\cdot|=\grd^{-1}(\cdot)$, the map

$\grd$ is a continuous map, thus $\leq$ is $\scrT$-consistent.
For a discretization of a subset $X$ of the plane with the associated face partial order $\le$ in  Fig. <ref>[left 1 and 2] the pre-order $\le$ is a discretization of the topology $\scrT$ of $X$. The map $\grd\colon X\twoheadrightarrow \cX$ assigns a vertex, edge or square to any point in $X$.  For closure algebras, $\disc$ is induced by sending a point $x$ to the cell in which it is contained, as in Eqn. (<ref>).
We can summarize these considerations as follows:

A surjective, continuous discretization map $\disc\colon (X,\scrT)\xtwoheadrightarrow{} (\cX,\le)$ is equivalent to a CA-discretization $(\cX,\cl,|\cdot|)$ with $(\sSet(\cX),\cl)$ dual[In terms of Boolean algebras with operators, cf. App. <ref> the pre-order is the dual to the closure algebra and vice versa. This duality can also be understood in terms of (co)-Heyting algebras.] to $(\cX,\le)$ and $|\cdot|=\disc^{-1}$.

If we allow the evaluation map $|\cdot|\colon \sSet(\cX) \to \sSet(X)$ to be an arbitrary homomorphism then the map $\disc$  defined
in (<ref>) is still valid

which allows us to treat the theory of discretization  with arbitrary, not necessarily surjective, continuous maps $\disc\colon X\to \cX$.
In this paper we restrict to  surjective discretization maps unless stated otherwise.

For a discretization map $\grd\colon X\twoheadrightarrow \cX$, there is always a $\scrT$-consistent pre-order.  Namely, the trivial, or indiscrete topology on $\cX$: $\cl\varnothing=\varnothing$ and $\cl~\cU = \cX$ for all $\cU\neq\varnothing$, i.e. $\le$ is an equivalence: $\xi\le \xi'$ and $\xi'\le \xi$ for all $\xi,\xi'\in \cX$.

The finest topology for which $\grd$ is continuous is given by the final topology on $\cX$ with respect to $\grd$.

For a pre-order
$(\cX,\le)$ we  define
a down-set $\cU\subset \cX$ by the property: $\xi'\in \cU$, $\xi\le \xi'$, then $\xi\in \cU$. The set of down-sets is denoted by
$\sO(\cX,\le)$[If there is no ambiguity about the pre-order we write $\sO(\cX)$ for short. Another common notation is $\Invsetpl(\le)$.]
which by construction  is  a finite distributive lattice with binary operations $\cap $ and $\cup$, cf. [16].
In a similar fashion we can define the lattice of up-sets $\sU(\cX,\le)$.

The join-irreducible elements of both lattices are characterized as

\[
\cl~\xi = \big\downarrow\xi := \setof{\xi'\mid \xi' \leq \xi} \quad\text{and}\quad \st \xi = \big\uparrow\xi := \setof{\xi'\mid \xi \leq \xi'},\quad \xi\in \cX,
\]

which are called the principal down-sets and up-sets respectively.
Intersections of up-sets and down-sets are the convex sets in $(\cX,\le)$[The convex sets in the pre-order $(\cX,\le)$  are exactly the locally closed subsets in $\cX$ as topological space.]

and are denoted by $\sCo(\cX,\le)$ which is a meet-semilattice with respect to $\cap$.

For a pre-order $(\cX,\le)$ we can define the partial equivalence classes by $\xi\sim \xi'$ if and only if
$\xi\le \xi'$ and $\xi'\le \xi$. The set of partial equivalence classes is denoted by $\cX/_\sim$. The latter is a poset via $[\xi]\le [\eta]$ if only only if $\xi\le \eta$. This yields the natural order-preserving projection
$\cX \xrightarrow{\pi} \cX/_\sim$ defined by $\xi \mapsto [\xi]$.
By construction $\sO(\cX,\le) \cong \sO(\cX/_\sim)$.
The map $X\twoheadrightarrow\cX\twoheadrightarrow\cX/_\sim$ is also a discretization.

The associated CA-discretization $\bigl(\cX/_\sim, \cl,|\cdot| \bigr)$ is defined by
$\cl [\xi] = \big\downarrow [\xi]$ and $\bigl|[\xi]\bigr| = \bigcup_{\xi'\in [\xi]}|\xi'|$ and yields a $T_0$ Alexandrov topology.

If $\le$ is a $\scrT$-consistent pre-order for a discretization map $\grd$, then by (<ref>) the down-sets $\cU$ for $\le$ correspond to the closed sets in $\cX$ and therefore by the continuity of $\grd$ we have that $\grd^{-1}(\cU) \in \scrC(X,\scrT)$,
where $\scrC(X,\scrT)$ denotes the closed sets in $X$.
This yields the lattice homomorphism

\begin{equation}
\label{spec2}
\grd^{-1}\colon \sO(\cX,\le) \longrightarrow \scrC(X,\scrT)
\xrightarrow[]{\subset} \sSet(X).
\end{equation}

In this setting we refer to $\grd^{-1}$ as an $\sO(\cX/_\sim)$-filtering on $X$ and  use the filtering notation:
F_\cU X := \grd^{-1}(\cU) = |\cU|.

Similarly, a subset $\cU\subset \cX$ is open if it is an up-set of $\cX$ and the image under $\grd^{-1}$ are open sets in $X$.
Dual to the $\sO(\cX/_\sim)$-filtering is the grading $X = \bigcup_{\xi\in [\xi]} G_{[\xi]} X$ with the property that $G_{[\xi]}X\neq \varnothing$ for all $\xi\in \cX$, cf. App. <ref>. The latter is an $\cX/_\sim$-grading given by $G_{[\xi]}X\xmapsto{\grade} [\xi]$, where $\bigl\{G_{[\xi]}X \mid[\xi]\in (\cX/_\sim,\le) \bigr\}$ is an ordered tessellation.
The following scheme shows the duality between CA-discritizations and continuous discretization maps, and between filterings and gradings, cf. App. <ref>:

\[
\begin{diagram}
\node{\displaystyle{\wbinom{(\cX,|\cdot|,\cl)}{ \text{CA-discretization}}}}\arrow{s,<>}\arrow[2]{e}\node{~}\node{\displaystyle{\wbinom{|\cdot|\colon \sO(\cX,\le) \to \sSet(X)}{ \text{filtering} }}}\arrow{s,<>}\\
\node{\displaystyle{\wbinom{\disc\colon (X,\scrT)\twoheadrightarrow (\cX,\le)}{\text{ continuous discretization}}}}\arrow[2]{e}\node{~}\node{\displaystyle{\wbinom{X = \bigcup_{[\xi]}G_{[\xi]} X}{\text{grading}}}}
\end{diagram}
% \begin{diagram}
% \node{\displaystyle{(\cX,|\cdot|,\cl) \atop \text{CA-discretization}}}\arrow{s,<>}\arrow[2]{e}\node{~}\node{\displaystyle{|\cdot|\colon \sO(\cX,\le) \to \sSet(X) \atop \text{filtering} }}\arrow{s,<>}\\
% \node{\displaystyle{\disc\colon (X,\scrT)\twoheadrightarrow (\cX,\le) \atop \text{ continuous discretization}}}\arrow[2]{e}\node{~}\node{\displaystyle{X = \bigcup_{[\xi]}G_{[\xi]} X \atop \text{grading}}}
% \end{diagram}
\]

where $G_{[\xi]}X=\disc^{-1}([\xi])$.
A grading on a topological space yields a discretization which is not necessarily continuous.[Let $X=\bigcup_{p\in \sP} G_pX$ be a grading of $X$ with embedding
$G_pX \xmapsto{\grade} X$.

Then, the map $\grade\colon X\to \sP$, defined by $\grade(x) =
\grade(G_pX)=p$ for all $x\in G_pX$, is a discretization map in the sense of Rmk. <ref>. By restricting to the range one obtains a surjective discretization map.]

A discretization map $\disc\colon X\twoheadrightarrow \cX$ is natural
if it is continuous and  the associated filtering $\disc^{-1}\colon \sO(\cX,\le)\to  \sSet(X)$ consists of mutually good pairs.[Recall that a pair $(X,A)$, $A\subset X$ closed, is a good pair  if
$A$ is a deformation retract of a neighborhood in $X$, q.v. <cit.>.]

A pre-order $\le$ is a $\scrT$-co-consistent with respect to $\grd$ if $\grd\colon X \to (\cX,\ge)$ is continuous, where $\ge $ is the opposite pre-order. This is equivalent to the condition
$\cl~ \grd^{-1}(\cU) \subset \grd^{-1}(\cl^* \cU)$, for all $\cU\subset \cX$,
where $\cl^*\cU:=\st \cU$ is the conjugate closure operator, cf. [46]. In terms of realization this reads $\cl|\cU|\subset |\st\cU|$.

If $\cU$ is $\cl^*$-closed, i.e. $\cl^*\cU = \st \cU = \cU$ which implies that $\cU$ is open in $(\cX,\le)$, then the $\scrT$-co-consistency implies that $|\cU|$ is a closed set.
Indeed, $\cl |\cU| \subset |\st\cU|=|\cU| \subset \cl |\cU|$ and thus $\cl|\cU|=|\cU|$. Moreover, the closed sets $\cU\in \sO(\cX,\le)$ for a $\scrT$-co-consistent pre-order are open under realization. If $\cU\in\sO(\cX,\le)$
then $\cl~\cU=\cU$ and therefore $\cU^c$ is open which implies that $\st\cU^c=\cU^c$. By the previous $|\cU^c| = |\cU|^c$ is closed and thus
$|\cU|$ is open.

§.§ Regular closed sets

Of particular importance in this paper are the  regular closed sets which play a central role in the construction of Morse tessellations. A subset $U\subset X$ is regular closed if $\cl\Int U = U$.
The set of regular closed sets in any topological space $X$ is denoted by $\scrR(X,\scrT)$ and the latter forms a complete Boolean algebra with unary operation $U^\#=\cl~U^c$    and binary operations $U\vee U' = U\cup U'$ and $U\wedge U' = \cl\Int(U\cap U')$, cf. <cit.>.

By the same token we can define the regular closed sets in a CA-discretization $(\cX,\cl,|\cdot|)$
and we denote regular closed sets in $\cX$ by $\scrR(\cX)$.[In this text we mainly consider regular closed sets with repect to one topology $\scrT$. Therefore the notation $\scrR(X)$ and $\scrR(\cX)$ does not cause any ambiguities.]

Regular closed subsets can be obtained from closed subsets.

The map $\cl\Int \colon \sO(\cX,\le) \to \scrR(\cX)$, defined by $\cU \mapsto \cl\Int \cU$, is a surjective lattice homomorphism.[This result holds for any topological space $X$.]

The atoms in $\sSet(\cX)$ are given by the set $\cX$. Since  $\scrR(\cX)$ is a finite Boolean algebra it is a power set on a set of atoms. Define the maximal elements in $\cX$ with respect to $\le$ by $\cX^{\topc}$, i.e. $\xi\in \cX^{\topc}$ if and only if $\st\xi = \xi$. Such elements are called top cells and form an anti-chain in $(\cX,\le)$.

The atoms of $\scrR(\cX)$ are given by the set $\bigl\{\cl\, \xi\mid \xi\in \cX^{\topc} \bigr\}$.
Moreover, $\cl\colon \sSet(\cX^{\topc}) \to \scrR(\cX)$ is an isomorphism with inverse $\cU\mapsto \cU\cap \cX^{\topc}$.

We start with the observation that $\xi\in \cX^{\topc}$ represents an open subset.
Indeed, $\st\xi=\xi$ and thus $\xi=\Int\xi$. Then, $\cl\,\xi = \cl\Int\xi$ is a regular closed set since $\cl\Int\cl\Int \xi = \cl\Int \xi$.
Suppose $\xi\not\in \cX^{\topc}$, i.e. $\xi$ is not maximal in $(\cX,\le)$.
Since $\xi=\{\xi\}$ is a singleton set and since $\Int\{\xi\}\subset\{\xi\}$ we conclude that $\Int\xi=\xi$, or $\Int\xi=\varnothing$. Suppose the former holds. Then, since $\xi$ is not maximal, $\{\xi\}\subsetneq \st \xi = \{\xi\}$, a contradiction.

Therefore, $\Int\xi=\varnothing$.
Let $\cU = \bigcup \{\xi\}$ be a regular closed set. Then,  by invoking Proposition <ref> we have

\[
\cU = \cl\Int\cU = \cl\Int\Bigl(\bigcup\{\xi\}\Bigr)
= \bigcup \cl\Int \xi =\bigcup\Bigl\{\cl\,\xi\mid \xi\in \cX^{\topc}\Bigr\},
\]

which proves that every regular closed set is union of elements $\cl\,\xi$, for $\xi\in \cX^{\topc}$. The latter form an anti-chain in $\scrR(\cX)$, i.e.,
again invoking Propositon <ref>,

\begin{equation}
\label{emptyforwed}
\cl\,\xi \wedge \cl\,\xi' = \cl\Int\xi \wedge \cl\Int\xi'  =
\cl\Int \bigl(  \xi \cap  \xi'\bigr) =\varnothing,
\end{equation}

which proves that $\cl\,\xi$, for $\xi\in \cX^{\topc}$, are atoms.
The map $\cl\colon \sSet(\cX^{\topc}) \to \scrR(\cX)$ preserves union. Consider $\cl~ \cU\wedge \cl~\cU'$. By (<ref>) we have

\[
\cl~ \cU\wedge \cl~\cU' = \bigcup\cl~\xi \wedge \bigcup\cl~\xi'
= \bigcup \cl~\xi'' = \cl \Bigl(\bigcup\{\xi''\} \Bigr),
\]

where $\bigcup\{\xi''\} = \cU\cap \cU'$, which proves that
$\cl\colon \sSet(\cX^{\topc}) \to \scrR(\cX)$ is a homomorphism.
Any $\cU\in \scrR(\cX)$ is uniquely represented as $\cU=\bigcup \cl~\xi=
\cl\bigl(\bigcup\{\xi\} \bigr)
$, $\xi\in \cX^\top$ which shows that $\cU\cap \cX^\top$ yields the unique set of generating cells $\xi\in \cX^\top$.

Regular closed sets in $\cX$ do not necessarily yield regular closed sets in $X$ under evaluation.
However, if $(\cX,\cl,|\cdot|)$ is Boolean, cf. Defn. <ref>, then we have the following correspondence:

$|\cU| = |\cl\Int \cU| = \cl|\Int\cU| = \cl\Int|\cU|$[If $(\cX,\cl,|\cdot|)$ is Boolean then $\Int|\cU| = |\Int\cU|$ follows from the relation for closure.] and thus a subset $\cU\subset\cX$ is regular closed in $\cX$ if and only if $|\cU|\subset X$ is regular closed in $X$.
This way the image of $|\cdot|\colon \scrR(\cX) \to \scrR(X)$

the finite  Boolean subalgebra $\scrR_0(X)$  contained in $\scrR(X)$. The subalgebra $\scrR_0(X)$ is  generated by the

set of atoms $\sJ\bigl( \scrR_0(X)\bigr) := \bigl\{\cl|\xi|\mid \xi\in \cX^{\topc}\bigr\}$.[The join-irreducible element in a finite Boolean algebra and the atoms that generate the Boolean algebra.]

Let $(\cX,\cl,|\cdot|)$ be a Boolean CA-discretization for $X$.

Then, the map

\[
\lbr\cdot\rbr\colon \sSet(\cX^{\topc}) \to \scrR_0(X),\quad \xi \mapsto \lbr\xi\rbr := \cl|\xi|,\footnote{The notation $\lbr\cdot\rbr$ is called \emph{closed realization}.}
\]

is a lattice isomorphism and thus a Boolean isomorphism.

Conversely, a finite sub-algebra $\scrR_0(X)\subset \scrR(X)$ yields a finite
closure algebra $\bigl(\sSet(\cX),\cl\bigr)$ as follows.
Define $[\scrR_0(X)]$ as the smallest sub-algebra  in $\sSet(X)$ containing $\scrR_0(X)$. Denote the set of atoms
by $\sJ\bigl( [\scrR_0(X)]\bigr) := \bigl\{|\xi| \mid \xi\in \cX\bigr\}$ for some finite set $\cX$.
This defines the pre-order $(\cX,\le)$ via the relation: $\xi\le \xi'$ if and only if $|\xi|\subset \cl|\xi'|$. Via the pre-order we obtain a closure operator on $\sSet(\cX)$ via (<ref>).

Let $\scrR_0(X)\subset \scrR(X)$ be a finite sub-algebra.
Then, $[\scrR_0(X)]\subset \sSet(X)$ defines a unique finite closure algebra
$\bigl(\sSet(\cX),\cl\bigr)$, where the $\cl$ is defined by (<ref>).

The above statement can be rephrased as:

a finite sub-algebra $\scrR_0(X)\subset \scrR(X)$  induces a unique Boolean CA-discretization $(\cX,\cl,|\cdot|)$  for $X$, where $|\cdot|$ an injective homomorphism of closure algebras.

§.§ CW-decomposition maps

Consider a  discretization
$\disc\colon X \twoheadrightarrow \cX$ without indicating a specific pre-order on $\cX$ for now.

Let  $B^q$ and $\bar B^q$

denote the open and closed unit balls in $\R^q$ respectively (where $B^0$ and $\bar{B}^0$ denote the one point space).  We say that $\xi$ is an $q$-cell if $|\xi|$ is homeomorphic to an open ball $B^q$.  The integer $q$ is called the dimension of $\xi$ and is denoted $\dim(\xi)$.
Suppose $\disc$ has the property that

every $\xi$ is  an $q$-cell for some $q$.

Given such a discretization map assigning dimension to a cell is an order-preserving map

\[
\dim\colon \cX \to (\N,\le),
\]

when $\cX$ is regarded as anti-chain.

Note therefore that the anti-chain $\cX$ is a naturally graded set with respect to $\dim$, i.e., $\cX=\bigcup_{q\in \N} G_q\cX$, where $G_q\cX=\dim^{-1}(q)$ is the set of $q$-cells. The latter also

yields the filtering

\[
\big\downarrow q \mapsto F_{\downarrow q}\cX,\quad \big\downarrow q \in \sO(\N),
%\varnothing \subset F_0 \cX \subset F_1\cX \subset \cdots \subset F_d\cX,
\]

where, by Birkhoff duality, $F_{\downarrow q}\cX = \dim^{-1}\left(\big\downarrow q\right)$, $\big\downarrow q =\{0,1,\cdots,q\}$ and $q\in \N$.
The composition $X\xrightarrow{\disc}\cX\xrightarrow{\dim}\N$, denoted by $\skel$,  also defines a discretization on $X$ and yields the filtering
$\big\downarrow q \mapsto F_{\downarrow q} X$, $\big\downarrow q \in \sO(\N)$.

A CW-decomposition map on $X$, denoted by $\cell\colon X\twoheadrightarrow\cX$, is a discretization map where each $\xi\in \cX$ is an $q$-cell for some $q$.  Moreover, for every $\xi\in \cX$ there is a continuous map $f_\xi\colon \bar{B}^q\to X$,  where $q=\dim(\xi)$, such that

(i) $f_\xi$ restricts to a homeomorphism $f_\xi|_{B^q}\colon B^q\to |\xi|$;

(ii) $f_\xi ( \bar{B}^q\smin B^q)\subset F_{\downarrow (q-1)}X$.

A subset $U\subset X$ is open (closed) if and only if    $f_\xi^{-1}(U)$ is open (closed) in $\bar B^q$
for all $\xi\in \cX$.

Note that if $X$ admits a (finite) CW-decomposition map then $X$ is a compact Hausdorff space, cf. [32].

Since $X$ is Hausdorff, it follows that $\lbr \xi\rbr = f_\xi(\bar{B}^q)$.[By continuity $f_\xi(\bar{B}^q) \subset \cl f_\xi({B}^q)= \lbr\xi\rbr$. On the other hand $\lbr \xi\rbr = \cl f_\xi({B}^q) \subset \cl f_\xi(\bar{B}^q) = f_\xi(\bar{B}^q)$ since the continuous image of a compact set is compact and in a Hausdorff space compact sets are closed.] CW-decompositions are general enough to include simplicial and cubical complexes.

From a CW-decomposition map we can define the following finite topology in terms of a pre-order on $\cX$:

\[
\xi\le \xi' \quad\text{if and only if}\quad |\xi|\subset \cl|\xi'|,
\]

which is called the face partial order on $\cX$. The name is justified by the following result.

The pre-order $(\cX,\le)$ is a partial order and the associated Alexandrov topology is
a $T_0$ topology.

Suppose $\xi\neq\xi'$ and $\xi\sim\xi'$, i.e. $\xi\le \xi'$ and $\xi'\le \xi$, which implies that $\cl|\xi|=\cl|\xi'|$ and thus $f_\xi(\bar B^q) = f_{\xi'}(\bar B^q)$.

$f_\xi(\bar B^q)  = f_\xi( \bar{B}^q\smin B^q \cup B^q) = f_\xi( \bar{B}^q\smin B^q) \cup |\xi|$,
and similarly $f_{\xi'}(\bar B^q) = f_{\xi'}( \bar{B}^q\smin B^q) \cup |\xi'|$.
By assumption $|\xi|\cap |\xi'|=\varnothing$, which yields that

\[
|\xi|\subset f_{\xi'}( \bar{B}^q\smin B^q) \subset F_{\downarrow (q-1)}X,\quad
|\xi'|\subset f_{\xi}( \bar{B}^q\smin B^q) \subset F_{\downarrow (q-1)}X.
\]

This contradicts the definition of $F_{\downarrow (q-1)}X$ and the fact that all cells are realized as disjoint sets in $X$. Therefore $\xi\sim \xi'$ if and only if $\xi=\xi'$ and $\le$ is a partial order and the Alexandrov topology is $T_0$.

The next step is to show that $\cell\colon X\to (\cX,\leq)$ is a discretization map, i.e., that $\le$ is $\scrT$-consistent.  Moreover, we also show that $\dim$ is order-preserving.

Let $\cl\colon \sSet(\cX) \to \sSet(\cX)$ be the closure operator defined by $\le$. Then,

(i) The face partial order $\leq$ is a $\scrT$-consistent.  Moreover, $\cl|\xi| = |\cl~\xi|$ for all $\xi\in \cX$;

(ii) $\dim\colon (\cX,\le) \to (\N,\le)$ is order-preserving.

We have that $\cl|\xi| = \lbr\xi\rbr = f_\xi(\bar B^q)$ and thus, as before using Definition <ref>(i)-(ii),
$f_\xi(\bar B^q)  = f_\xi( \bar{B}^q\smin B^q \cup B^q) = f_\xi( \bar{B}^q\smin B^q) \cup |\xi|$.
This implies that $\cl|\xi|$ is a union of sets $|\xi'|$ and more precisely

\[
\cl|\xi| = \bigcup \bigl\{ |\xi'|\mid |\xi'|\subset \cl|\xi|\bigr\}
=\bigcup \bigl\{ |\xi'|\mid \xi'\le \xi\bigr\} = \bigl|\cl~\xi\bigr|,
\]

which proves (i).
As for (ii) we have that $\cl|\xi| \smin |\xi|=f_\xi( \bar{B}^q\smin B^q) = \bigcup \bigl\{ |\xi'|\mid \xi'\le \xi\bigr\}\subset F_{\downarrow (q-1)}X$
and thus $\dim(\xi')\le q-1$. Consequently, $\xi'\le \xi$ implies $\dim(\xi')\le \dim(\xi)$.

Lemma <ref> shows in particular that $\cell$ is continuous open map and thus a discretization with respect to the face partial order. Moreover, $|\cdot|$ is an injective homomorphism of closure algebras. Indeed by additivity of $|\cdot|$ and $\cl$ we have that
$\cl|\cU| = |\cl~\cU|$ for all $\cU\in \sSet(\cX)$.
The associated CA-discretization $(\cX,\cl,|\cdot|)$ is Boolean and is called a CW-decomposition for $X$.

A CW-decomposition map $\cell\colon X\to \cX$ is a natural discretization map.[cf. Defn. <ref>.]

For every  closed subset $\cU\subset \cX$ the realization $|\cU| \subset X$ is a sub CW-decomposition and therefore a deformation retract of a neighborhood in $X$, cf. <cit.>.

§ BI-TOPOLOGICAL SPACES AND DISCRETIZATION

A triple $(X,\scrT,\scrT')$ is called a bi-topological space if the factors $(X,\scrT)$ and $(X,\scrT')$ are well-defined topological spaces.
The associated closure algebra for $(X,\scrT,\scrT')$ is the Boolean algebra with operators $(\sSet(X),\cl,\cl')$, where $\cl$ and $\cl'$ are the closure operators for $\scrT$ and $\scrT'$ respectively and is referred to as bi-closure algebra for  $(X,\scrT,\scrT')$.

A subset $U\subset X$ is a $(\scrT,\scrT')$-pairwise clopen set for $X$ if
$U$ is closed in $\scrT$ and open in $\scrT'$.  We denote the set of $(\scrT,\scrT')$-pairwise clopen sets in a bi-topological space $(X,\scrT,\scrT')$ by $\aclop(X)$.
Similarly, we can define $(\scrT',\scrT)$-pairwise clopen sets which are
open in $\scrT$ and closed in $\scrT'$ and are denoted by

The next step is to consider discretization for bi-topological spaces.

A discretization map $\disc\colon X\twoheadrightarrow \cX$ is a p-continuous map[J.C. Kelly refers to such map that are continuous with respect to both topologies as p-continuous, or pairwise continuous maps, cf. [41].]  between bi-topological spaces if there exists pre-orders $(\cX,\le)$ and $(\cX,\le')$ that are $\scrT$-consistent and $\scrT'$-consistent respectively. We write
$\disc\colon (X,\scrT,\scrT') \twoheadrightarrow (\cX,\le,\le')$. The associated bi-topological CA-discretization is denoted by $(\cX,\cl,\cl',|\cdot|)$ where $\cl$ and $\cl'$ are the associated closure operators.

Let $(\cX,\cl,\cl',|\cdot|)$ be a bi-topological CA-discretization for $(X,\scrT,\scrT')$. This is equivalent
to choosing a discretization map $\disc\colon X\twoheadrightarrow \cX$ and pre-orders $(\cX,\le)$ and $(\cX,\le')$ such that $\disc$ is continuous with respect to both $\scrT$ and $\scrT'$.

Since the pre-orders $\le$ and $\le'$ represent Alexandrov topologies we can coarsen the finite topologies using both up-sets and down-sets.

Let $\disc\colon X\twoheadrightarrow \cX$ be a discretization map.
An antagonistic pre-order for $(X,\scrT,\scrT')$ is a pre-order $(\cX,\le^\dagger)$ such that

(i) $\le^\dagger$ is $\scrT$-consistent with respect to $\grd$;

(ii) $\le^\dagger$  is $\scrT'$-co-consistent with respect to $\grd$.

These conditions   translate as

\begin{equation}
\label{dubbleconst}
\cl|\cU|\subset | \cl^\dagger\cU|,\quad \cl'|\cU| \subset |\st^\dagger\cU|,\quad \forall \cU\subset \cX,
\end{equation}

where $\st^\dagger = (\cl^\dagger)^*$, the conjugate closure operator, cf. Sect. <ref> and [34].
The triple $(\cX,\cl^\dagger,|\cdot|)$ is called an antagonistic CA-discretization for $(X,\scrT,\scrT')$.

Antagonistic pre-orders can also be defined by reversing the role of $\scrT$ and $\scrT'$.

An equivalent way to say that $(\cX,\le^\dagger)$ is an antagonistic pre-order is that both

\[
\disc\colon (X,\scrT) \xtwoheadrightarrow{} (\cX,\le^\dagger),\quad\text{and}\quad \disc\colon (X,\scrT') \xtwoheadrightarrow{} (\cX,\ge^\dagger),
\]

are continuous.

Let $(\cX,\le^\dagger)$ be an antagonistic pre-order as in Definition
Closed sets $\cU$ in $(\cX,\le^\dagger)$ are pairwise clopen sets $|\cU|$ in $\aclop(X)$, cf. Sect. <ref>.

Alternatively, if $(\cX,\cl,\cl',|\cdot|)$ is a bi-topological CA discretization for $(X,\scrT,\scrT')$ then the lattice embedding:

\begin{equation}
\label{dubbleconst2}
\iota\colon\sAnt \rightarrowtail \sO(\cX,\le) \cap \sU(\cX,\le') =: \aclop(\cX),
\end{equation}

yields an antagonistic pre-order for $(X,\scrT,\scrT')$.
if we use Birkhoff duality to dualize the homomorphism

$\iota\colon\sAnt \rightarrowtail\sSet(\cX)$, q.v. Thm. <ref> and Rmk. <ref>, we obtain the order-preserving surjection

\begin{equation}
\label{defndyn}
\begin{aligned}
\pi\colon \cX \twoheadrightarrow \sP,\quad \xi \mapsto \pi(\xi) &:= \min\bigl\{ p\in \sP\mid \xi\in \iota\bigl(\big\downarrow p\bigr)\bigr\}\\
&~=\max\Bigl\{\min\bigl\{ \cU\in \sJ(\sAnt)\mid \xi\in \cU\bigr\}\Bigr\},
\end{aligned}
% \sJ(\iota)\colon \cX \twoheadrightarrow \sJ\bigl( \sAnt\bigr),\quad \xi \mapsto \sJ(\iota)(\xi) := \min\bigl\{ \cU\in \sAnt\mid \xi\in \cU\bigr\}
\end{equation}

where $\sJ\bigl( \sAnt\bigr)$ is the   poset of join-irreducible elements in $\sAnt$.

By construction  a pre-order $(\cX,\le^\dagger)$ is defined by

\[
\xi\le^\dagger \xi' \quad\hbox{if and only if}\quad \pi(\xi) \le \pi(\xi'),
% \xi\le^\dagger \xi' \quad\hbox{if and only if}\quad \sJ(\iota)(\xi) \subseteq \sJ(\iota)(\xi'),
\]

is an antagonistic pre-order with $\sO(\cX,\le^\dagger)\cong \sAnt$, cf. App. <ref>.
In this case we say that $\le^\dagger$ is
an antagonistic coarsening of $(\cX,\cl,\cl',|\cdot|)$.
The embedding $\sO(\cX,\le^\dagger) \rightarrowtail\sSet(\cX)$ is dual to the identity map

\begin{equation}
\label{idred}
\id\colon \cX \to (\cX,\le^\dagger).
\end{equation}

A pre-order $\le^\dagger$ is an
antagonistic pre-order for $\disc\colon (X,\scrT,\scrT') \to (\cX,\le^\dagger)$
if and only if there exists a bi-topological CA-discretization $(\cX,\cl,\cl',|\cdot|)$ such
that $\sO(\cX,\le^\dagger)$ is given by
(<ref>), i.e. an antagonistic pre-order is equivalent to an antagonistic coarsening.

One direction is given be the construction in (<ref>).
It remains to show that an antagonistic pre-order satisfying Definition <ref>(i)-(ii) is an antagonistic coarsening. If we define the discrete closure operators  $\cl = \cl^\dagger$ and $\cl'=\st^\dagger$, then
$\sO(\cX,\le) = \sO(\cX,\le^\dagger)$ and $\sU(\cX,\le') = \sO(\cX,\le^\dagger)$, which proves the theorem.

In  practical situations we can choose $\sO(\cX,\le^\dagger) = \aclop(\cX)$.
Let $\sJ(\sO(\cX,\le^\dagger))$ be the poset of join-irreducible elements in $\aclop(\cX)$. From the results in [40, 36] we consider a representation $(\sSC,\le)$ of
$\sJ(\sO(\cX,\le^\dagger))$ defined by

\begin{equation}
\label{defnofSC}
\sSC := \bigl\{ \cS = \cU\smin\cU^\pred\mid \cU\in \sJ(\sO(\cX,\le^\dagger))\bigr\},\footnote{For every $\cU\in \sJ(\sO(\cX,\le^\dagger))$ there exists a unique immediate predecessor $\cU^\pred\in  \sO(\cX,\le^\dagger)$.}
\end{equation}

$\cS\le \cS'$ if and only if $\cU\subset \cU'$, with $\cU,\cU'\in \sJ(\sO(\cX,\le^\dagger))$ uniquely
determined by $\cS= \cU\smin\cU^\pred$ and $\cS' = \cU'\smin\cU'^\pred$.
If we regard the poset $(\cX,\le^\dagger)$ as a directed graph then
the sets $\cS\in \sSC$ correspond to the strongly connected components of the directed graph which is the motivation for the
abbreviation $\sSC$.
The posets
$(\sSC,\le)$ and $\bigl(\sJ(\sO(\cX,\le^\dagger)),\subset \bigr)$ are isomorphic by construction and
the elements in $\sSC$ correspond to the partial equivalence classes in $(\cX,\le^\dagger)$, cf. Rmk. <ref>, which yields the (order-preserving) projection $\dyn\colon(\cX,\le^\dagger) \twoheadrightarrow (\sSC,\le) = \cX/_\sim$.[We use the terminology $\dyn$ as it is used later on in the setting of the .] The latter may be regarded as  a finite discretization of $\cX$.

The embedding
$\sO(\cX,\le^\dagger) \xrightarrow{\subset}\sO(\cX,\le)$ implies that
$\dyn$ is also an order-preserving map $\dyn\colon (\cX,\le)\twoheadrightarrow (\sSC,\le)$ which factors as
$(\cX,\le) \xtwoheadrightarrow{\id} (\cX,\le^\dagger)
\xtwoheadrightarrow{\dyn} (\sSC,\le)$. Indeed, a downset in $(\cX,\le^\dagger)$ is a downset in $(\cX,\le)$ and therefore $\le\, \subset \,\le^\dagger$ as pre-orders.

By the same token we have that $\part$ is order-reversing with respect to $\le'$ which follows from the embedding $\sO(\cX,\le^\dagger) \xrightarrow{\subset}\sU(\cX,\le')$.
Summarizing, the maps $\part$

\begin{equation}
\label{ordpreforgrd}
\begin{diagram}
\node{(\cX,\le)}\arrow{s,l}{\id}\arrow{see,l}{\dyn}\node{}\node{}\\
\node{(\cX,\le^\dagger)}\arrow[2]{e,l}{\dyn}\node{}\node{\sSC~~}\\
\node{(\cX,\le')}\arrow{n,l,--}{\id}\arrow{nee,r,--}{\dyn}\node{}\node{}
\end{diagram}
\end{equation}

are order-preserving and order-reversing respectively.

The map $\dyn$ is again a continuous discretization and the composition

\begin{equation}
\label{spacegrading}
X\xtwoheadrightarrow{\disc} \cX \xtwoheadrightarrow{\dyn} \sSC
%X\xrightarrow{ \disc} \cX \xrightarrow{\part} \sSC,
\end{equation}

denoted by $\tile$,
may be regarded as a continuous map $\tile\colon (X,\scrT) \to (\sSC,\le)$ by factoring through $(\cX,\le)$
and as a continuous map  $(X,\scrT') \to (\sSC,\ge)$ by factoring through $(\cX,\le')$.

Every antagonistic pre-order for $(X,\scrT,\scrT')$ defines a grading $G_\cS X\xmapsto{\tile}\cS$ of $X$ given by

\[
X = \bigcup_{\cS\in \sSC} G_\cS X, \quad G_\cS X =
\bigl(\tile^{-1}\bigr)(\cS),
% \bigl(\disc^{-1}\circ\skel^{-1}\bigr)(\cS),
\]

which is called an antagonistic tessellation of $X$. We apply these bi-topological discretization techniques for the in the next two sections.

An antagonistic pre-order $(\cX,\le^\dagger)$ satisfies $\le\, \subset \,\le^\dagger$ and $\le'\, \subset \,\le^\dagger$ as pre-orders
and thus $\le\vee\le'\,\subset \,\le^\dagger$ as pre-orders.
The `vee' on the pre-order is the transitive closure of the union.
The case $\sO(\cX,\le^\dagger) = \aclop(\cX)$ corresponds to $\le^\dagger \,=\, \le\vee\le'$.

§ FLOW TOPOLOGIES AND DISCRETIZATION OF DYNAMICS

In the previous sections we discussed
topological and bi-topological spaces in terms of closure algebras
which is the appropriate language for formalizing discretization of topology.

The next step
is to model the dynamics of
semi-flows on topological spaces

via  appropriately constructed topologies on $X$.

Such topologies may be realized in many different ways and we refer to the dynamics induced topologies as flow topologies.
The objective of this paper is not to develop explicit methods for discretizing dynamics but to describe the contours of a theory that discretizes dynamics in terms of discretizing two topologies.

§.§ Dynamics as topology

As discussed in the previous sections a topological space $(X,\scrT)$  can be equivalently described via the closure algebra $\cl\colon\sSet(X) \to \sSet(X)$.

For a semi-flow $\phi$ define the modal operators[cf. Sect. <ref>, Axioms (M1)-(M2).] $\der^-, \der^+\colon \sSet(X)\to \sSet(X)$  given by

\begin{equation}
\label{secondcl3}
U \mapsto \der^- U := \bigcup_{t> 0} \phi(-t,U),\quad U \mapsto \der^+ U := \bigcup_{t> 0} \phi(t,U),
\end{equation}

which are called the strict backward image and strict forward image operators respectively.
The operators $\cl^-=\id\cup \der^-$ and $\cl^+ = \id \cup \der^+$ are obtained by taking $t\ge 0$ and  satisfy

all four Kuratowski axioms (K1)-(K4) for closure operators.

The closure operators $\der^-$ and $\der^+$ define the topologies $\scrT^-$ and $\scrT^+$ on $X$ respectively, which are Alexandrov topologies on $X$.
The associated specialization pre-order on $X$ defined by $\scrT^+$ will   be denoted by $\le^+$ and is
defined by $y\le^+ x$ if and only if $y\in \rmGamma^+\{x\}$. The latter is
characterized by

\[
y\le^+ x\quad\text{if and only if}\quad y = \phi(t,x)\quad\text{for some}\quad t\ge 0.
\]

The pre-order $\le^+$ does record the directionality of the flow $\phi$ but discards the sense of time and invariance.
The closed sets in $\scrT^+$ are the forward invariant sets for $\phi$ and are denoted by $\Invsetpl(\phi)$.
The two topologies $\scrT$ and $\scrT^+$ combined comprise the bi-topological space $(X,\scrT,\scrT^+)$.
The associated specialization pre-order on $X$ defined by $\scrT^-$ will   be denoted by $\le^-$ and is
defined by $y\le^- x$ if and only if $y\in \rmGamma^-\{x\}$. This pre-order is characterized by
$y\le^- x$ if and only if $y \in\phi(-t,x)$ for some $t\ge 0$, i.e. $x=\phi(t,y)$ for some $t\ge 0$. This shows that $\le^-$ is the opposite pre-order to $\le^+$
and closed sets in $\scrT^+$ are open sets in $\scrT^-$ and vice versa.
The Alexandrov topologies $\scrT^-$ and $\scrT^+$ are each other's opposites.
The closed sets in $\scrT^-$ are the backward invariant sets for $\phi$ and are denoted by $\Invsetneg(\phi)$.
The two topologies $\scrT$ and $\scrT^-$ combined comprise the bi-topological space $(X,\scrT,\scrT^-)$.

For the topologies $\scrT^-$ and $\scrT^+$  the closure  and conjugate closure operators are related: $(\cl^+)^* = \st^+= \cl^-$ and $(\cl^-)^* = \st^- = \cl^+$.

Since the topologies $\scrT^-$ and  $\scrT^+$ are Alexandrov the associated specialization pre-orders $\le^-$ and  $\le^+$ equivalently describe the topologies.
Therefore one can also think of $(X,\scrT,\le^-)$ and $(X,\scrT,\le^+)$ as a pospaces, i.e  $(X,\scrT)$ is a topological space  while $(X,\le^-)$ and $(X,\le^+)$ are pre-orders, cf. [60].

Since $\cl^-$ and $\cl^+$ are closure operators $\der^-$ and $\der^+$ are
(canonical) derivative operators satisfying  the Axioms (D1)-(D3).

Observe that

\begin{equation}
\begin{aligned}
\label{strongdop144}
\der^+\bigl(\der^+ U\bigr)  &= \bigcup_{t>0} \phi\Bigl(t,\bigcup_{s>0} \phi(s,U)\Bigr) = \bigcup_{t>0} \bigcup_{s>0}\phi(t,\phi(s,U))\\
&= \bigcup_{s+t >0} \phi(s+t,U) = \der^+ U,
\end{aligned}
\end{equation}

and therefore
$\der^+$ satisfies the stronger idempotency axiom  (K3), i.e. $\der^+(\der^+ U) = \der^+ U$ and $\der^+$ is an idempotent derivative operator.
In the same way one proves that $\der^-$ satisfies the idempotency axiom in (K3).

The above derivative operator is associated with a  binary relation on $X$:
$y<^+ x$ if and only if $y = \phi(t,x)$ for some $t> 0$.

Observe that the $<^+$ is a transitive relation and the reflexive closure yields the specialization pre-order $\le^+$.
Points $x\in X$ for which  $\der^+ \{x\} =\{x\}$ correspond to fixed of $\phi$ and are examples of reflexive point for $<^+$, i.e. $x<^+ x$.
Other reflexive points are given by periodic orbits for $\phi$, cf. [1].
The derivative operator $\der^+$ does not  detects invariant sets in general. Indeed, for $\phi(t,x) = xe^t$ the set $U=(0,\infty)$ satisfies $\der^+ U=U$ but is not invariant. To capture invariance one can use $\tau$-forward image operator $\Gammatau$, cf. Sect. <ref>.

For the definition of the Alexandrov topology $\scrT^+$ continuity of $\phi$ with respect to the topology $\scrT$ is not required and therefore the topologies are independently defined.
Continuity of $\phi(t,\cdot)$ with respect to the topology $\scrT^+$ is immediate.
Continuity of $\phi$ with respect to $\scrT$ yields interaction of the two topologies as a manifestation of the semi-flow $\phi$ on $X$.

If $\phi$ is continuous in the $\scrT$-topology then we have the following  necessary condition: $\rmGamma^+\cl ~U \subset \cl~ \rmGamma^+U$ for all $U\subset X$.
by the continuity of $\phi$ we have that $\phi(t,\cl~U) \subset \cl~\phi(t,U)$ for all $U\subset X$ and for all $t\ge 0$.

\begin{equation}
\label{subcomm12}
\begin{aligned}
\cl~\der^+ U &= \cl~\bigcup_{t > 0}\phi(t,U) \supset \bigcup_{t> 0}\cl~\phi(t,U)
\supset \bigcup_{t> 0}\phi(t,\cl~U)\\
&= \der^+\cl~U.
\end{aligned}
\end{equation}

This implies the subcommutativity for $\scrT$ and $\scrT^+$, i.e.
$U\subset X$ $\scrT^+$-closed implies that $\cl~U$ is $\scrT^+$-closed. This
makes the space $(X,\scrT,\scrT^+)$ a $(\scrT,\scrT^+)$-subcommutative bi-topological space.

For the opposite topology the same applies. As in (<ref>) we have that $\der^-\Int U \subset \Int \der^- U$, which implies $U\subset X$ $\scrT^-$-closed implies that $\Int U$ is $\scrT^-$-closed.
We again refer to $(X,\scrT,\scrT^-)$ as a  $(\scrT,\scrT^-)$-subcommutative bi-topological space.

§.§ The $\underline\scrT^0$

As our aim is an algebraization of dynamics that recovers invariance
based on Wazewski's principle, cf.  [13], we study the

attracting and repelling blocks of $\phi$.
Recall that closed attracting blocks are defined by

\begin{equation}
\label{clattbl12}
\sABlockC(\phi) := \bigl\{ U\subset X\mid \cl~U=U,~~\phi(t,U) \subset \Int U,~~\forall t>0\bigr\}.
\end{equation}

Let $U,U'\in \sABlockC(\phi)$.
If the singular homology satisfies

$H(U,U')\neq 0$, then $\Inv(U\smin U') \neq \varnothing$, cf. Sect. <ref>.

In order incorporate attracting blocks into the theory of (bi)-closure algebras we construct derived topologies. In general, there are a number of options to define derived topologies. We highlight one particular choice that serves the purpose of constructing closed attracting blocks, cf. Sect. <ref>.

The operator $\uderm:=\der^-\cl$ is a modal operator.[A modal operator on $\sSet(X)$ is operator which is additive and preserves the empty set, cf. App. <ref>.]

Axioms (M1)-(M2) in App. <ref> are satisfied since $\cl$  is a closure operator and $\der^-$ is a derivative operator,

which establishes $\uderm$ as a normal, additive operator on $\sSet(X)$.

The operator $\uderm$ does not define a derivative operator since (D3) is not satisfied in general.
We can still use $\uderm$ to define a topology on $X$.

A subset $U\subset X$ is $\underline\scrT^0$-closed if and only if $\uderm U \subset U$.

It remains to show that $\underline\scrT^0$-closed sets indeed define a topology on $X$. Observe that the condition that $U\subset X$ is $\underline\scrT^0$-closed is equivalent to the condition $\phi(-t,\cl~U)\subset U$ for all $t>0$.

Let $U,U'\subset X$ be $\underline\scrT^0$-closed. Then,
$U\cup U'$ is $\underline\scrT^0$-closed.

By assumption

\[
\begin{aligned}
\phi\bigl(-t,\cl(U\cup U')\bigr) &=\phi\bigl(-t,\cl~U\cup \cl~U')\bigr)
= \phi(-t,\cl~U)\cup \phi(-t,\cl~U')\\
&\subset U\cup U',~~\forall t>0.
\end{aligned}
\]

Therefore, $\uderm(U\cup U') \subset U\cup U'$, which proves that $U\cup U'$ is $\underline\scrT^0$-closed.

Let $\{U_i\}_{i\in I}$ an arbitrary collection of $\underline\scrT^0$-closed sets in $X$. Then, the intersection $\bigcap_{i\in I} U_i$
is $\underline\scrT^0$-closed.

For the closure of $\bigcap_{i\in I} U_i$ it holds that $\cl\bigl( \bigcap_{i\in I} U_i\bigr) \subset \bigcap_{i\in I} \cl~U_i$.
By assumption

\[
\begin{aligned}
\phi\Bigl(-t,\cl\Bigl( \bigcap_{i\in I} U_i\Bigr)\Bigr) &\subset \phi\Bigl(-t,\bigcap_{i\in I} \cl~U_i\Bigr)
= \bigcap_{i\in I} \phi(-t,\cl~U_i)\\
&\subset \bigcap_{i\in I} U_i,~~\forall t>0.
\end{aligned}
\]

Therefore, $\uderm\Bigl(\bigcap_{i\in I} U_i\Bigr) \subset \bigcap_{i\in I} U_i$,
which proves that $\bigcap_{i\in I} U_i$ is $\underline\scrT^0$-closed.

By Lemmas <ref> and <ref> the $\underline\scrT^0$-closed sets  comprise the  $\underline\scrT^0$. The associated closure operator is given by

\begin{equation}
\label{udermtop3}
\ucl^0 U := \bigcap\bigl\{U\subset U'\mid U' ~~\text{$\underline\scrT^0$-closed} \bigr\}.
\end{equation}

By definition   the $\underline\scrT^0$ is not necessarily an Alexandrov topology.
A subset $U\subset X$ is $\underline\scrT^0$-closed if $\ucl^0 U = U$.

The $\underline\scrT^0$ is a coarsening of the Alexandrov topology $\scrT^-$ which is generated by $\cl^-$, i.e. $\cl^- \subset \ucl^0$.

By definition $\der^- U \subset \uderm U\subset U$ which implies $\der^- U\subset U$. Therefore,
$\cl^- \subset \ucl^0$.

The  maps $\phi(t,\cdot)$ are continuous in the $\underline\scrT^0$ for all $t\ge 0$.

For the backward image operator we have that $\phi(-t,\der^- U) = \der^-\phi(-t,U)$, $t\ge 0$, and thus $\phi(-t,\der^-\cl~ U) = \der^-\phi(-t,\cl~ U) \supset \der^-\cl~\phi(-t, U)$.
Suppose $U$ is $\underline\scrT^0$-closed, i.e. $\uderm U\subset U$.

\[
\uderm \phi(-t,U)\subset \phi(-t,\uderm U) \subset \phi(-t,U),
\]

which proves that $\phi(-t,U)$ is $\underline\scrT^0$-closed.
The latter proves that for every $t\ge 0$ the map $\phi(t,\cdot)$ is continuous in the $\underline\scrT^0$-flow topology.

Observe, by Proposition <ref>, that $\phi(t,\cdot)$ is continuous in both topologies $\scrT$ and $\underline\scrT^0$.
The $\underline\scrT^0$

defines a new, derived subcommutative bi-topological space $(X,\scrT,\underline\scrT^0)$. The latter is not a pospace in general.
Open and closed sets in the $\underline\scrT^0$  can be characterized via the semi-flow $\phi$.

A subset $U\subset X$ is closed in the $\underline\scrT^0$ if and only if

\begin{equation}
\label{attblock14bc}
\phi(-t,\cl~U)\subset U,\quad \forall t>0.
\end{equation}

Similarly, a subset $U\subset X$ is open in the $\underline\scrT^0$ if and only if

\begin{equation}
\label{attblock15bc}
\phi(t,U)\subset \Int U,\quad \forall t>0.
\end{equation}

By definition a subset $U\subset X$ is $\underline\scrT^0$-closed if and only if $\udero U\subset U$
which implies $\uderm U \subset U$ and thus $\der^- \cl~U\subset U$. The latter implies that $\phi(-t,\cl~U)\subset U$ for all $t>0$.
Conversely, if (<ref>) holds then $\der^- \cl~U\subset U$ and thus
$\udero U \subset U$.

By definition a subset $U\subset X$ is $\underline\scrT^0$-open if and only if $U^c$ is $\underline\scrT^0$-closed, i.e.
$\phi(-t,\cl~U^c) \subset U^c$ for all $t>0$. The latter is equivalent to $\phi(-t,\cl~U^c)^c \supset U$ for all $t>0$, which results in the equivalent statement that $\phi(-t,\Int U) \supset U$ for all $t>0$.
If we compose the latter with $\phi(t,\cdot)$ we obtain

\[
\phi(t,U) \subset \phi\bigr(t,\phi(-t,\Int U) \bigr)\subset \Int U,\quad\forall t>0.
\]

On the hand,  composition of  (<ref>) with $\phi(-t,\cdot)$ gives
$U \subset \phi\bigl(-t,\phi(t,U) \bigr) \subset \Int U$, for all $t>0$, which prove that $U$ is $\underline\scrT^0$-open.

The $(\scrT,\underline\scrT^0)$-pairwise  clopen sets in $(X,\scrT,\underline\scrT^0)$ are
defined as sets $U\subset X$ that are closed in $\scrT$ and open in $\underline\scrT^0$.

A subset $U\subset X$ is a
closed attracting block, cf. (<ref>),
if and only if $U$ is a $(\scrT,\underline\scrT^0)$-pairwise  clopen set in $(X,\scrT,\underline\scrT^0)$.

This follows from Lemma <ref> and the definition of closed attracting blocks.

set of   closed attracting blocks $\sABlockC(\phi)$ is
a sublattice of $\sSet(X)$, cf. [38, 39], [36].
This lattice is not complete in general.

The continuity of $\phi$ in the $\scrT$-topology
can be relaxed to $\R^+$-parameter family $\phi(t,\cdot)$ of  continuous in the $\scrT$-topology for all $t\ge 0$.
This implies in particular that we can apply discretization and topologization other families of maps such as $t\in \Z^+$ which is equivalent to iterating a map, i.e. discrete time dynamics. Continuity of $\phi(t,\cdot)$
implies a more equal role for both topologies.
The continuity of $\phi$ on $\R^+\times X$ comes in in two instances, (i) equivalent discretization via discrete resolutions, cf. Sect. <ref>, and (ii) algebraization in order to invoke Wazewski's principle for finding invariant sets, cf. Sect. <ref> and [13].

We can use the pairwise clopen sets in $(X,\scrT,\underline\scrT^0)$ as a base (of closed sets) for the topology $\scrT^\dagger$.
This yields the continuous discretization map $\disc\colon (X,\scrT^\dagger)\twoheadrightarrow (\cX,\le^\dagger)$. We say that $\scrT^\dagger$ is an antagonistic topology with respect to the pair $(\scrT,\underline\scrT^0)$.

If we consider variations of the operator $\uderm$ such as $\uder = \der^+\cl$  then sets that are open in $(X,\scrT)$ and closed in $(X,\underline\scrT^0)$ are open attracting blocks. Other variations entail $\cl~\der^+$ and $\cl~\der^-$, cf. Rem. <ref> and Sect. <ref>. Our definition of is suitable for the theory in this paper.

§.§ Discretization of the $\underline\scrT^0$-topology

In Section <ref> we discussed discretization of topology in terms of closure algebras and derivative algebras. The standard CW-decompositions of  spaces are examples of such
discretizations. In this section we apply the closure algebra discretization to the which provides the appropriate discretization of dynamics.

As we have encoded $\phi$ as a topological space $(X,\scrT,\underline\scrT^0)$, we can apply the tools from Section <ref>, e.g., CA-discretizations, DA-discretizations, discretization maps and topology consistent pre-orders.

Let  $\ucl^0\colon \sSet(\cX) \to \sSet(\cX)$ be a closure
such that
$\ucl^0 |\cU| \subset |\ucl^0 \cU|$ for all $\cU\in \sSet(\cX)$.
This induces a pre-order $\le^0$ by (<ref>) and yields the continuity of $\grd\colon X \to (\cX,\le^0)$, cf. Rmk. <ref>.
Conversely, if $\le^0$ is any $\underline\scrT^0$-consistent pre-order with  respect to $\grd$ then the associated
closure operator $\ucl^0$ defines a CA-discretization for $(X,\underline\scrT^0)$.

A pre-order $\le^0$ on $\cX$ with associated closure operator $\ucl^0\colon \sSet(\cX) \to \sSet(\cX)$ is $\underline\scrT^0$-consistent with respect to $\disc\colon X \twoheadrightarrow\cX$ if and only if

\begin{equation}
\label{secondtop6}
\phi\bigl(-t,\cl |\xi|\bigr) \subset \bigl| \ucl^0\xi\bigr|,\quad
\forall t > 0,
% \forall t \ge 0,
\quad\text{and}\quad \forall \xi\in \cX.
\end{equation}

The discretization $\disc\colon X\twoheadrightarrow \cX$ is continuous if and only if $\cU$ closed in $(\cX,\le^0)$ implies that $|\cU|$ is $\underline\scrT^0$-closed.
Assume (<ref>) is satisfied. Let $\ucl^0 \cU = \cU$, then

\[
\begin{aligned}
\phi(-t,\cl|\cU|) &= \bigcup_{\xi\in \cU} \phi(-t,\cl|\xi|)\subset \bigcup_{\xi\in \cU}|\ucl^0\xi|\\
&= \Bigl|\ucl^0\Bigl(\bigcup_{\xi\in\cU}\{\xi\}\Bigl)\Bigr| = |\ucl^0\cU| =|\cU|,\quad\forall t>0,
\end{aligned}
\]

which proves that $\uderm|\cU| \subset |\cU|$ and thus $|\cU|$ is $\underline\scrT^0$-closed.

Conversely, if $\grd$ is continuous,
then $\cU$ closed in $(\cX,\le^0)$ implies $|\cU|$ is $\underline\scrT^0$-closed and
therefore $\phi(-t,\cl|\cU|)\subset |\cU|$. Choose $\cU = \ucl^0\xi$. This implies that

\[
\phi(-t,\cl|\xi|) \subset \phi(-t,\cl|\ucl^0\xi|)\subset |\ucl^0\xi|,\quad\forall t>0,\quad\text{and}\quad \forall \xi\in \cX.
\]

which establishes (<ref>).

In the discrete setting we can define the discrete analogues of the $(X,\underline\scrT^0)$ via discretizations of $\scrT$ and $\scrT^-$.
Let $\der^-\colon \sSet(\cX) \to \sSet(\cX)$ a discrete  derivative for $\scrT^-$ and define the additive operator $\uderm:= \der^-\cl\colon \sSet(\cX) \to \sSet(\cX)$.

The operator
$\udero\colon \sSet(\cX)\to \sSet(\cX)$, defined by

\begin{equation}
\label{discderfs1}
\udero \cU := \bigl(\uderm \bigr)^{\bm +}\cU,
\footnote{The transitive closure $\mathrm\Theta^{\bm +}$ for any modal operator $\mathrm\Theta$ on $\sSet(\cX)$ is defined in the same way.}
\quad \hbox{with}\quad  \bigl(\uderm \bigr)^{\bm +} := \bigcup_{k=1}^\infty \bigl( \uderm \bigr)^k, \quad \cU\subset \cX,
\end{equation}

is a discrete derivative operator and the
operator $\ucl^0 := \id \cup\udero$ defines a closure operator on $\sSet(\cX)$.
The triple $(\cX,\ucl^0,|\cdot|)$
is a CA-discretization for $(X,\underline\scrT^0)$.

To show that the operator $\udero$ is a derivative we first show that it is sub-idempotent. Observe that since $\cX$ is finite the definition of $\udero$ holds for a union up to some finite $k=k_*$. Indeed, there are only finitely many sets $\bigl(\uderm \bigr)^{k}(\cU)$ for finitely many subsets $\cU$.

\[
\bigl(\uderm \bigr)^\ell\udero\cU = \bigcup_{k=1+\ell}^{k_*+\ell} \bigl( \uderm \bigr)^{k}\cU
\]

which uses the (finite) additivity of $\uderm$.
This implies $\udero\udero \cU \subset \udero\cU$ and thus $\udero$ is a (strong) derivative on $\sSet(\cX)$.

It remains to show that $\disc\colon X\twoheadrightarrow\cX$ is continuous.
We have, using the CA-discretization $(\cX,\cl,|\cdot|)$ for $(X,\scrT)$
and the DA-discretization $(\cX,\der^-,|\cdot|)$ for $(X,\scrT)$, that

\[
\uderm |\xi| = \der^-\cl|\xi| \subset \der^-|\cl\xi|\subset |\der^-\cl~\xi|
= |\uderm\xi|.
\]

This implies $\phi(-t,\cl|\xi|) \subset \uderm |\xi| \subset |\uderm \xi| \subset |\udero \xi| \subset |\ucl^0\xi|$ for all $t>0$ which, by Lemma <ref>, proves that $\disc$ continuous.

The following lemma  formulates a criterion
for discretizing the flow topology $\underline\scrT^0$ without using a discretization for $\scrT^-$.

Let $\uderup\colon \sSet(\cX) \to \sSet(\cX)$ be a modal
operator such that

\begin{equation}
\label{secondtop5}
\phi\bigl(-t,\cl|\xi|\bigr) \subset \bigl| \uderup\xi\bigr|,\quad \forall t> 0,\quad\text{and}\quad \forall \xi\in \cX.
\end{equation}

Then, the  operator   $\uderup^{\bm +}\colon \sSet(\cX)\to \sSet(\cX)$  is a derivative operator. The closure operator $\ucl^0:=\id \cup \uderup^{\bm +}$ yields a  CA-discretization of the $(X,\underline\scrT^0)$.

The fact that $\uderup^{\bm+}$ is sub-idempotent and thus a derivative operator follows from the construction in the proof of Lemma <ref>.
By assumption $\bigcup_{t>0} \phi(-t,\cl |\xi|) = \uderm |\xi| \subset |\uderup \xi|$.
As in the proof of Lemma <ref> this implies that
$\phi(-t,\cl|\xi|) \subset \uderm |\xi| \subset |\uderup \xi| \subset |\uderup^{\bm +} \xi| \subset |\ucl^0\xi|$
for all $t>0$ which proves that $\disc$ is continuous, completing the proof.

The following result gives a local version of the above criterion and provides a practical method for constructing CA-discretizations for the . Figure <ref> gives an example of a discretization of both $(X,\scrT)$ and $(X,\underline\scrT^0)$.

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The topological space $(X,\scrT)$ and semi-flow $\phi$ [left 1].  A discretization of $X$ with $\scrT$-consistent pre-order $(\cX,\le)$  [middle 2, 3].
The relation $\uderup^{-1}$ [middle 4] which generates the
$\underline\scrT^0$-consistent pre-order $(\cX,\le^0)$ [right 5].
The discretization map $\disc\colon X\to\cX$ is continuous with respect to both topologies. Common coarsening of $(\cX,\le)$ and $(\cX,\ge^0)$ [right 6] resulting in a Morse pre-order $(\cX,\le^\dagger)$
[right 7].   All pre-orders are represented by their Hasse diagrams.

Let $(X,\scrT)$ be compact and let $\uderup\colon \sSet(\cX) \to \sSet(\cX)$ be a modal operator.
Assume that for every $\xi\in \cX$ there exists $t_\xi>0$ such that

\begin{equation}
\label{secondtop555}
\phi\bigl(-t,\cl|\xi|\bigr) \subset \bigl| \uderup\xi\bigr|,\quad \forall 0<t\le t_\xi,\quad\text{and}\quad \forall \xi\in \cX.
\end{equation}

Then, $(\cX,\ucl^0,|\cdot|)$, with $\ucl^0= \mathrm\Theta^{\bm{+=}}=\id\cup\uderup^{\bm +}$, is a CA-discretization for  $(X,\underline\scrT^0)$.

The proof of based on the following observation. By the compactness of $X$ we may assume, without loss of generality, that $t_\xi\ge t_*>0$ for all $\xi\in \cX$.
Then, $\phi(-t,\cl|\xi|) \subset |\uderup\xi|$ for all $0<t\le t_*$ and for all $\xi\in \cX$.
Observe that

\[
\begin{aligned}
\phi(-2t,\cl|\xi|) &= \phi\bigl(-t, \phi(-t,\cl|\xi|)\bigr)\subset \phi\bigl(-t,|\uderup\xi|\bigr)\\
&\subset  \phi\bigl(-t,\cl|\uderup \xi|\bigr)\subset | \uderup^2\xi|,
\end{aligned}
\]

which yields $\phi(-kt,\cl|\xi|)\subset  | \uderup^k\xi|
\subset |\uderup^{\bm +}\cU|$,  for all $k\ge 1$ and for all $0<t\le t_*$. As  in the proof of Lemma <ref>,
$\phi(-t,\cl|\xi|)  \subset |\uderup^{\bm +} \xi| \subset |\ucl^0\xi|$
for all $t>0$ which proves that $\disc$ is continuous,
and thus $(\cX,\id\cup\uderup^{\bm +},|\cdot|)$ is a CA-discretization for  $(X,\underline\scrT^0)$.

The operator $\uderup$ defines a relation on $\cX$: $(\eta,\xi) \in \uderup^{-1}$ if and only if $\eta\in \uderup \xi$, cf. App. <ref>. The transitive, reflexive closure of $\uderup^{-1}$ is the pre-order $\le^0$ associated to
$\ucl^0 := \id\cup\uderup^{\bm +}$.

For discretizing the flow topologies $\scrT^-$ and $\scrT^+$ we can use the criteria in Lemmas <ref> and <ref> by discarding the topology $\scrT$, i.e. take $\cl$ to be the identity map. For example a  discrete closure operator $\cl^+\colon\sSet(\cX) \to \sSet(\cX)$ yields a CA-discretization for $\scrT^+$ if and only if $\phi(t,|\xi|)\subset |\cl^+\xi|$ for all $t\ge 0$ and for all $\xi\in \cX$, i.e. $\scrT^+$-consistency for the associated pre-order $\le^+$.

The definition of the in Section <ref> uses the modal operator $\uderm$. This construction works for any modal operator $\mathrm\Theta$ on $\sSet(X)$ and can be used for variations on the .

§ MORSE PRE-ORDERS AND DISCRETE RESOLUTIONS

In this section we explain the implications of discretization with respect to two topologies in the sense of $(\scrT,\underline\scrT^0)$- pairwise clopen sets.

We consider the derived bi-topological space $(X,\scrT,\underline\scrT^0)$.
Let $\disc\colon X\twoheadrightarrow \cX$ be a discretization map and let
$(\cX,\le^\dagger)$ be an anatagonistic pre-order
for $(X,\scrT,\underline\scrT^0)$, which motivates the following definition:

Let $\disc\colon X\twoheadrightarrow \cX$ be a discretization map.
A Morse pre-order on $\cX$ is an anatagonistic pre-order $\le^\dagger$
for $(X,\scrT,\underline\scrT^0)$, i.e.

(i) $\le^\dagger$ is $\scrT$-consistent with respect to $\grd$;

(ii) $\le^\dagger$  is $\underline\scrT^0$-co-consistent with respect to $\grd$.

The associated closure operator
is denoted by $\cl^\dagger\colon \sSet(\cX)\to \sSet(\cX)$.

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(1) – (e);
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Intersection of both the lattice of down-sets $\sO(\cX,\leq)$ and up-sets $\sU(\cX,\le^0)$ for Fig. <ref> yields the coarsening $(\sSC,\leq)$ [right] of the Morse pre-order $(\cX,\le^\dagger)$.

By Theorem <ref> $(\cX,\le^\dagger)$ is an antagonistic coarsening of discretizations for both
$\scrT$ and $\underline\scrT^0$, cf. Sect. <ref>.

The conditions for an antagonistic pre-order imply that for $\cU\subset \cX$ we have that (i) $\cl|\cU| \subset |\cl^\dagger\cU|$ and (ii) $\ucl^0|\cU| \subset |\st^\dagger\cU|$. In particular, if $\cU\in \sO(\cX,\le^\dagger)$, then $\cl^\dagger\cU=\cU$ and thus  $|\cU|$ is $\scrT$-closed. Moreover,
if $\cU\in \sO(\cX,\le^\dagger)$, then
$|\cU|$ is $\underline\scrT^0$-open, cf. Sect. <ref>, which implies that $|\cU|$ satisfies

$\phi(t,|\cU|) \subset \Int |\cU|$ for $t>0$, cf. Lemma <ref>.
These facts combined show that $\cU\in \sO(\cX,\le^\dagger)$ implies that $|\cU|$
is a $(\scrT,\underline\scrT^0)$-pairwise clopen set and therefore a closed attracting block, i.e.
$|\cU|\in \sABlockC(\phi)$.
We have the following commutative diagram:

\begin{equation}
\dgARROWLENGTH=2.0em
\label{AB12}
\begin{diagram}
\node{\sSet(X)} \node{\sABlockC(\phi)}\arrow{w,l,V}{\supset}\\
\node{\sSet(\cX)}\arrow{n,l,V}{|\cdot|} \node{\sO(\cX,\le^\dagger)}\arrow{w,l,V}{\supset}\arrow{n,r,V}{|\cdot|}
\end{diagram}
\end{equation}

The $(\underline\scrT^0,\scrT)$-pairwise clopen sets correspond to open repelling blocks. Indeed, $|\cU|$ is open and (<ref>) we have that $\phi(-t,\cl~|\cU|) \subset |\cU|$ for all $t>0$. We denote the open repelling blocks by $\sRBlockO(\phi)$.

§.§ Morse tessellations

For a Morse pre-order $(\cX,\le^\dagger)$ the down-sets $\sO(\cX,\le^\dagger)$ yield a sublattice of closed attracting blocks.

Following the theory in Section <ref>
a Morse  pre-order $(\cX,\leq^\dagger)$ yields a finite discretization of $\cX$ that we denote by  $\dyn\colon\cX \twoheadrightarrow
\sSC$ and which is defined by combining the formulas in (<ref>) and (<ref>).
The latter is dual to the embedding $\sO(\sSC) \cong\sO(\cX,\le^\dagger)\rightarrowtail \sSet(\cX)$.

The composition

\[
X\xtwoheadrightarrow{ \grd} \cX \xtwoheadrightarrow{\dyn} \sSC,
\]

defines a continuous $T_0$-discretization of $X$ which is denoted by $\tile\colon X\to \sSC$,
cf. (<ref>) and which defines an $\sSC$-grading on $X$, cf. App. <ref>.
Compare the latter with the composition $X\xtwoheadrightarrow{\disc}\cX\xtwoheadrightarrow{\dim}\N$ which defines an $\N$-grading on $X$.

The diagrams in (<ref>) and (<ref>)  show how $\dyn$ is order-preserving and order-reversing with respect to $(\cX,\le)$ and $(\cX,\le^0)$ respectively, cf. Sect. <ref>.
Diagram (<ref>) also shows the continuous $T_0$-discretization maps

\[
\tile\colon (X,\scrT) \to (\sSC,\le) \quad \hbox{and}\quad \tile\colon (X,\underline\scrT^0) \to (\sSC^*,\ge),
\]

by factoring through $(\cX,\le)$
and  through $(\cX,\le^0)$ respectively.
A Morse pre-order can therefore be thought of as grading of $X$ which, as $T_0$-discretization, is a continuous map $\tile$ in the above sense.

A down-set $\cU$ in $(\sSC,\le)$ is a down-set in $(\cX,\le)$ and an up-set in $(\cX,\le^0)$ and therefore closed  in $(X,\scrT)$ and and open in $(X,\underline\scrT^0)$ respectively.
By Lemma <ref> this implies that down-set $\cU$ in $(\sSC,\le)$ realize to closed attracting neighborhoods $\tile^{-1}(\cU)\in \sABlockC(\phi)$.
On the level of classes $\cS\in \sSC$ the inverse image of $\tile$ yields an $\sSC$-graded tessellation
$(\sT,\le)$ with

\begin{equation}
\label{Morsetessll11}
\sT:= \bigl\{ T = \tile^{-1}(\cS)\mid \cS \in \sSC \bigr\},
\end{equation}

such that $\big\downarrow T$ is $\scrT$-closed and
$\underline\scrT^0$-open, i.e. $\phi(t,x)\in \Int \big\downarrow T$ for every $x\in T$ and for all tiles $T$. The latter follows since $\big\downarrow T = \big\downarrow\tile^{-1}(\cS) = \tile^{-1}\bigl(\big\downarrow \cS \bigr) $ and $\big\downarrow\cS$ is a down-set in $\sSC$,
cf. <cit.>.
This motivates the  definition:

An ordered tessellation $(\sT,\le)$  of $X$, cf. Defn. <ref>,

is called a Morse tessellation for $\phi$ if  for every $T\in \sT$

(i) $\big\downarrow T$ is $\scrT$-closed;

(ii) $\phi\bigl(t,x\bigr) \in \Int \big\downarrow T$, for all $x\in T$ and for all $t>0$, i.e.  $\big\downarrow T$ is $\underline\scrT^0$-open.

The sets $T\in \sT$ are called Morse tiles.[Equivalently, for every $I\in \sO(\sT,\le)$, $|I|:=\bigcup_{T\in I} T \in \sABlock(\phi)$, i.e. $\sT=\sT(\sN)$ where $\sN$ is given by  $\sN=\{|I|\mid I\in\sO(\sT,\le)\}\subset \sABlock(\phi)$.]

Conversely, Morse tessellations yield Morse pre-orders and associated space discretizations.
Indeed, for a Morse tessellation $(\sT,\le)$ we declare the tiles to be the cells in $\cX$ and the partial order is the Morse pre-order on $\cX$. By definition this defines a discretization for both $(X,\scrT)$ and $(X,\underline\scrT^0)$. In the next subsection we discuss a more refined reconstruction based on regular closed sets.

One can obviously build larger sets $\cX$ by for example considering $\scrT$ or $\underline\scrT^0$ closure of the tiles $T$. One can also define fine structure within the tiles. In the next section we explain a specific reconstruction in the case of regular closed tiles.

Morse tessellations are a defining structure for Morse representations, cf. [40]. The considerations
in this subsection explain that Morse tessellations are equivalent to Morse pre-orders which will be the central structure for discussing connection matrix theory in Sect. <ref>.

§.§ Regular closed attracting blocks

In this section
we study a special property of closed attracting blocks.

Let $U\in \sABlockC(\phi)$ be a closed attracting block, then $U$ is a regular closed attracting block, i.e. $\sABlockC(\phi) = \sABlockR(\phi)$, where the latter denotes the lattice of
regular closed attracting blocks. Moreover,

\[
U\wedge U' = U\cap U',
\]

for all $U,U'\in \sABlockC(\phi)$.

Let $U\subset X$ be a closed attracting block. By definition
$\cl \Int U  \subset  U$.
Suppose $U \smin \cl \Int U\neq\varnothing$.
Let $x\in  U \smin \cl \Int U$.
Choose $t_n>0$ with $t_n \to 0$ as $n \to \infty$. Since $U$ is an attracting block we have that $y_n := \phi(t_n,x) \in \Int U$ for all $t_n>0$.
By the continuity $y_n \to x$ as $n\to \infty$ which implies  that $x\in \cl \Int U$, a contradiction. Therefore
$ U=\cl \Int U$, which proves that $U$ is a regular closed attracting block, cf. [39, 40].

If $U,U'\in \sABlockC(\phi)$, then $U\cap U'\in \sABlockC(\phi)$ and thus $U\cap U'$ is a closed attracting block and therefore a regular closed attracting block. This implies that $U\cap U' =
\cl \Int(U\cap U') = U\wedge U'$.

By Theorem <ref> we may assume that a Morse pre-order $(\cX,\le^\dagger)$
is induced by a bi-topological CA-discretization $(\cX,\cl,\ucl^0,|\cdot|)$  for $(X,\scrT,\underline\scrT^0)$.
Assume without loss of generality that $\sO(\cX,\le^\dagger) = \aclop(\cX)$ and
$\cl = \cl^\dagger$ and $\ucl^0 = \st^\dagger$.

Moreover, assume that
$(\cX,\cl,|\cdot|)$ is Boolean CA-discretization.

Let $\cU\in \aclop(\cX)$, then, since $|\cU| \in \sABlockR(\phi)$, the set $\cU$ is regular closed and
$|\cU\cap \cU'|=|\cU|\cap|\cU'| = |\cU|\wedge|\cU'| = |\cU\wedge\cU'|$,
yields the following regular closed analogue of  (<ref>):

\begin{equation}
\label{resolution1}
\begin{diagram}
\node{\scrR(X)} \node{\sABlockR(\phi)}\arrow{w,l,V}{\supset}\\
\node{\scrR(\cX)}\arrow{n,l,V}{|\cdot|} \node{\sO(\cX,\le^\dagger)}\arrow{w,l,V}{\supset}\arrow{n,r,V}{|\cdot|}
\end{diagram}
% \quad\quad\quad
%     \begin{diagram}
%     \node{\scrR(X)} \node{\sABlockR(\phi)}\arrow{w,l,V}{\supset}\\
%     \node{\sSet(\cX^{\topc})}\arrow{n,l,V}{\lbr\cdot\rbr} \node{\sO\left(\sT\right)}\arrow{w,l,V}{}\arrow{n,r,V}{}
%     \end{diagram}
\end{equation}

The down-sets for a Morse pre-order yield a sublattice of $\sABlockC(\phi)$.
Conversely, for a finite sublattice  $\sN\subset \sABlockC(\phi)$ we can construct a Boolean
CA-discretization, cf. Sect. <ref> and Rmk. <ref>.
Morse pre-orders

provide an important reduction of discretization data as is explained in the next section.

§.§ Discrete resolutions

In this section we assume that
$(\cX,\cl,|\cdot|)$ is Boolean CA-discretization of $(X,\scrT)$.
The fact that closed attracting blocks are regular closed sets, cf. Thm. <ref>, is a crucial property for reducing the data structures in the theory of Morse pre-orders.
The bottom embedding $\sO(\cX,\le^\dagger) \rightarrowtail \scrR(\cX)$ in  (<ref>) is inclusion
since $\cU\cap \cU' = \cU\wedge \cU'$, which follows from the fact that $|\cdot|\colon\scrR(\cX)\rightarrowtail \scrR(X)$ is a homomorphism and $(\cX,\cl,|\cdot|)$ is Boolean.

Since $\scrR(\cX) \cong \sSet(\cX^{\topc})$, cf.  Prop. <ref>,

we can dualize the   homomorphism

\[
\sO(\sSC,\le) \xrightarrow{\cong}\sO(\cX,\le^\dagger) \rightarrowtail \scrR(\cX) \xrightarrow{\cong}\sSet(\cX^{\topc}).
\]

This yields  the surjection $\pi\colon\cX^{\topc} \twoheadrightarrow (\sSC,\le)$, where $\cX^\topc$ is unordered and where $\sSC \cong \sJ\bigl( \sO(\cX,\le^\dagger)\bigr)$.

induced pre-order $\le^\topc$ on $\cX^{\topc}$, defined by

\[
\xi \le^\topc\xi'  \quad \text{if and only if}
\quad \pi(\xi) \le \pi(\xi'),
\]

is called a called a discrete resolution for $\phi$.[Since  $\sO(\cX,\le^\dagger)\cong \sO(\sSC,\le)\cong \sO(\cX^\topc,\le^\topc)$ it follows that pre-order $\le^\topc$ is the restriction of $\le^\dagger$ to $\cX^\topc$, cf. Proof of Thm. <ref>. ]

By construction we have that $\sO(\cX,\le^\dagger) \cong \sO(\cX^\topc,\le^\topc)\cong \sO(\sSC,\le)$.

The associated closure operator on $\sSet(\cX^{\topc})$ is denoted by $\cl^\topc$.

By Proposition <ref>, $\cU^\topc\in \sO(\cX^\topc,\le^\topc)$ implies $\cl~\cU^{\topc} \in \sO(\cX,\le^\dagger)$ and thus, since $(\cX,\cl,|\cdot|)$ is Boolean,

$\cl~|\cU^\topc|=|\cl~\cU^\topc| \in \sABlockR(\phi)$. Consequently $\phi\bigl(t,|\cl~\cU^\topc|\bigr)\subset \Int |\cl~\cU^\topc|$ for all $t>0$.
Let $\cU^{\topc} = \cl^\topc\xi$, with $\xi\in \cX^{\topc}$. Then,
$\phi\bigl(t,\lbr\xi\rbr  \bigr) \subset \phi\bigl(t,|\cl~\cl^\topc\xi|\bigr)\subset \Int|\cl~\cl^\topc\xi| = \Int \cl|\cl^\topc\xi|
=\Int\lbr\cl^\topc\xi\rbr$ for all $t>0$.
The latter is a condition on only the top cells. We show now that any pre-order $(\cX^\topc,\le^\topc)$ satisfying the latter is
a discrete resolution induced by a Morse pre-order.

A pre-order $(\cX^\topc,\le^\topc)$ is a discrete resolution

if and only if

\begin{equation}
\label{firstchar1}
\phi\bigl(t,\lbr\xi\rbr\bigr) \subset
%\Int \Bigl\lbr\bigcup_{n\ge 0} \bvtheta_{\cF}^n[\xi]\Bigr\rbr
\Int \bigl\lbr \cl^\topc\xi\bigr\rbr,\quad \forall t>0.
\end{equation}

Condition (<ref>) is a characterization of discrete resolution and can be used as alternative definition of discrete resolution.

If $(X,\scrT)$ is a compact topological space  then
the criterion in (<ref>) is equivalent  to the condition: for every $\xi\in \cX^{\topc}$ there exists a $t_\xi>0$, such that
$\phi\bigl(t,\lbr\xi\rbr\bigr) \subset \Int \bigl\lbr\cl^\topc\xi %\bvtheta_{\cF^{\scaleto{+=}{4pt}}}[\xi]
\bigr\rbr$ for all
$0< t \le t_\xi$.

Any binary relation $\cF\subset \cX^\topc\times\cX^\topc$ for which $\cF^{\bm{+=}}$  satisfies (<ref>)

will also be referred to as a discrete resolution for $\phi$.
The definition of discrete resolution is reminiscent of the notion of weak outer approximation in relation to the commutative diagram in (<ref>), cf. [36], <cit.>, [40].

The Boolean algebra $\sSet(\cX^\topc)$ is a sublattice in $\sSet(\cX)$. The embedding does not preserve the top element and the inclusion is not therefore Boolean.
The closure operator
\cl\colon \sSet(\cX^\topc) \to \sSet(\cX)
$, given by $\cU \mapsto \cl~ \cU \in \sO(\cX,\le)$,
is additive but not a lattice homomorphism in general.

Since $i\colon\cX^\topc\hookrightarrow (\cX,\le)$, with $\cX^\topc$ unordered, is an  order-embedding[The top cells $\cX^\top$ form an anti-chain in $(\cX,\le)$, cf. Sect. <ref>.] Birkhoff duality yields the surjective lattice homomorphism $j\colon \sO(\cX,\le) \twoheadrightarrow \sSet(\cX^\topc)$ given by $\cU \mapsto \cU^\topc:= \cU\cap \cX^\topc$.

Schematically we pose the following lifting diagram:

\begin{equation}
\label{lift1}
\begin{diagram}
\node{}\node{\sO(\cX,\le)}\arrow{s,r,A}{j}\\
\node{\sO(\cX^\topc,\le^\topc)}\arrow{ne,l,..}{\cl}\arrow{e,l,V}{\id}\node{\sSet(\cX^\topc)}
\end{diagram}
\end{equation}

Theorem <ref> below shows that the identity map can  be lifted as  closure.

Suppose $(\cX^\topc,\le^\topc)$ is a pre-order that satisfies (<ref>).

Then, the restriction $\cl\colon \sO(\cX^\topc,\le^\topc)\rightarrowtail  \sO(\cX,\le)$ is an injective lattice homomorphism with $j\circ \cl = \id$.

By definition $\cU^{\topc} = \bigcup_{\xi\in \cU^{\topc}} \{\xi\}$ and since $\lbr\cdot\rbr\colon \sSet(\cX^{\topc}) \to \scrR(X)$
is an injective Boolean homomorphism, cf. Prop. <ref>, we have that
$\lbr\cU^{\topc}\rbr = \bigcup_{\xi\in \cU^{\topc}} \lbr\xi\rbr$. In combination with (<ref>), the additivity of $\cl^\topc$ and the fact that $\cU^{\topc}\in\sO(\cX^\topc,\le^\topc)$ we conclude

\[
\begin{aligned}
\phi\bigl(t,\lbr\cU^{\topc}\rbr\bigr) &= \bigcup_{\xi\in \cU^{\topc}} \phi\bigl(t,\lbr\xi\rbr\bigr)
\subset \bigcup_{\xi\in \cU^{\topc}}\Bigl( \Int\bigl\lbr \cl^\topc\xi \bigr\rbr\Bigr)
\subset \Int\Bigl(\bigcup_{\xi\in \cU^{\topc}} \bigl\lbr \cl^\topc\xi \bigr\rbr\Bigr)\\
&=\Int \Bigl\lbr\bigcup_{\xi\in \cU^{\topc}} \cl^\topc\xi \Bigr\rbr =\Int \bigl\lbr\cl^\topc \cU^{\topc}\bigr\rbr\subset \Int \lbr \cU^{\topc}\rbr,\quad \forall t>0,
\end{aligned}
\]

which proves that $\lbr\cU^{\topc}\rbr$ is an attracting block for $\phi$. Since $\sO(\cX^\topc,\le^\topc)$ and $\sABlockR(\phi)$ are sublattices of $\sSet(\cX^{\topc})$ and $\scrR(X)$ respectively, and since
$\lbr\cdot\rbr\colon \sSet(\cX^{\topc}) \rightarrowtail \scrR(X)$ is an injective Boolean homomorphism we  conclude
that the map $\lbr\cdot\rbr\colon \sO(\cX^\topc,\le^\topc) \rightarrowtail \sABlockR(\phi)$ is a injective lattice homomorphism. In particular we conclude that $|\cl~\cU^\topc|
\in \sABlockR(\phi)$ and $\cl~\cU^\topc \in \sO(\cX,\le)$.

To show that the restriction of $\cl$ is a homomorphism it remains to check that the unit and intersection are preserved.
By definition $\cl~ \cX^\topc = \cX$ which proves that the unit is preserved.
By Proposition <ref> we have that $\cl~\cU^\topc\in  %\sO(\cX,\le^\dagger)\rightarrowtail
\scrR(\cX)$ and by
Theorem <ref>, $|\cl~\cU^\topc|\cap |\cl~\cU'^\topc| =
|\cl~\cU^\topc|\wedge |\cl~\cU'^\topc|$.
This implies,

\[
\begin{aligned}
|\cl(\cU^\topc\cap \cU'^\topc)| &= \cl|\cU^\topc\cap \cU'^\topc| = \lbr \cU^\topc\cap \cU'^\topc\rbr = \lbr\cU^\topc\rbr \wedge \lbr \cU'^\topc\rbr = \lbr\cU^\topc\rbr \cap \lbr \cU'^\topc\rbr\\
&= |\cl~ \cU^\topc|\cap |\cl~ \cU'^\topc| = |\cl~\cU^\topc \cap \cl~ \cU'^\topc|,
\end{aligned}
\]

where we use Proposition <ref> to conclude that
$\lbr \cU^\topc\cap \cU'^\topc\rbr = \lbr\cU^\topc\rbr \wedge \lbr \cU'^\topc\rbr$.
The fact that $|\cdot|$ is injective yields   $\cl(\cU^\topc\cap \cU'^\topc) = \cl~\cU^\topc\cap \cl~\cU'^\topc$,  which completes the proof.

The direction that a discrete resolution satisfies (<ref>) is given above. It remains to show that (<ref>) yields a Morse pre-order.
Suppose (<ref>) is satisfied. Then, by Theorem <ref>, $\cl\colon \sO(\cX^\topc,\le^\topc) \to\sO(\cX,\le)$ provided an embedding sublattice.
From (<ref>)-(<ref>) we obtain a pre-order $(\cX,\le^\dagger)$ such that the range of the above closure is
the lattice $\sO(\cX,\le^\dagger)$:

\[
\begin{diagram}
\node{\sO(\cX,\le^\dagger)}\arrow{se,r,V}{\subset}\node{}\node{\sO(\cX^{\topc},\le^\topc)}\arrow[2]{w,lr}{\cl}{\cong}
\arrow{sw,r,V}{\cl}\\
\node{}\node{\sO(\cX,\le)}
\end{diagram}
\]

The pre-order $(\cX,\le^\dagger)$ is the desired Morse pre-order that induces $(\cX^\topc,\le^\topc)$.
If we choose $\le^0 \,= \,\ge^\dagger$ we obtain $\le^\dagger$ as antagonistic pre-order for $(\cX,\cl,\ucl^0,|\cdot|)$,
cf. Thm. <ref>.

The novelty of the above construction is that
the (injective) composition

\[
\sO(\sSC,\le)\xrightarrow{\cong}\sO(\cX^\topc,\le^\topc) \Rightarrowtail{\cl} \sO(\cX,\le) \Rightarrowtail{\subset} \sSet(\cX),
\]

dualizes to
the finite discretization

\begin{equation}
\label{dyndefn12}
\dyn\colon (\cX,\le) \xtwoheadrightarrow{} (\sSC,\le), %\cong \bigl(\cX^{\topc}/_\sim,\le^\topc\bigr),
\end{equation}

which is defined in Section <ref> and is given by the formulas in (<ref>) and (<ref>).

The finite discretization $\dyn$ recovers the Morse pre-order $\le^\dagger$ via $\xi\le^\dagger\xi'$
if and only if $\dyn(\xi)\le\dyn(\xi')$,
and $\le^\topc$ is the restriction of  $\le^\dagger$ to $\cX^{\topc}$.
The advantage of using regular closed  sets is that the Morse pre-order is completely determined by the restriction $(\cX^\topc,\le^\topc)$ which is a much smaller data structure in general and bypasses the  topologies given by
$\le$ and $\le^0$.
The following result gives a formula for determining $\dyn$ in terms of $\le^\topc$:

Suppose $(\cX^\topc,\le^\topc)$ is a pre-order that satisfies (<ref>).
The finite discretization $\dyn\colon (\cX,\le) \xtwoheadrightarrow{} (\sSC,\le)$ is given by

\begin{equation}
\label{thethmdynform}
\xi \mapsto \dyn(\xi)  = \min_{\sSC} \Bigl\{ \bigl[\eta^\topc\bigr]\mid \eta^\topc\in \st \xi \cap \cX^\topc \Bigr\},
\end{equation}

where $\bigl[\eta^\topc\bigr] \in \sSC$ is the partial equivalence class in $(\cX^\topc,\le^\topc)$ containing $\eta^\topc$.

Consider the commutative diagram

\begin{equation}\label{dyndiapr12}
\begin{diagram}
\node{\cX}\arrow{s,l}{\iota_\cX} \arrow{e,l,V}{\dyn}\node{\sSC}\arrow{s,r}{\iota_\sSC}\\
\node{\sJ(\sO(\cX,\le))} \arrow{e,l,V}{\sJ(\cl)}\node{\sJ(\sO(\sSC))}
\end{diagram}
\end{equation}

where the maps $\iota_\cX$ and $\iota_\sSC$ are given by $\xi \xmapsto{\iota_\cX} \big\downarrow \xi$ and
$[\xi^\topc] \xmapsto{\iota_\sSC} \big\downarrow [\xi^\topc]$, and $\cl\colon \sO(\sSC)
\rightarrowtail\sO(\cX,\le)$.
By the commutativity we have that $\dyn = \iota_\sSC^{-1}\circ \sJ(\cl)\circ \iota_\cX$, and $\sJ(\cl)(\cU) = \min \cl^{-1}\bigl(\big\uparrow \cU\bigr)\in \sJ(\sO(\sSC))$, $\cU\in \sJ(\sO(\cX,\le))$, cf. Thm. <ref>.
Recall that $\cl^{-1}(\cU) =\{\cU^\topc\in \sO(\sSC)\mid \cl~\cU^\topc =\cU\}$.
Note that in $\sO(\cX,\le)$ the up-set $\big\uparrow \iota_\cX(\xi)$  is the set of closed subsets in $\cX$ that contain $\iota_\cX(\xi)=\big\downarrow \xi$.
By definition $\cl^{-1}\bigl(\big\uparrow\iota_\cX(\xi) \bigr)$ are all $\cU^\topc\in \sO(\sSC)$ such that $\cl~\cU^\topc =\cU$ for some $\cU\in \sO(\cX,\le)$ with $\big\downarrow \xi\subset \cU$. The latter is equivalent to $\xi\in \cU$. Since $\cl~\cU^\topc\in \sO(\sSC)$ this implies that $\cl^{-1}\bigl(\big\uparrow\iota_\cX(\xi) \bigr) = \{ \cU^\topc\in \sO(\sSC)\mid \xi\in \cl~\cU^\topc\}$. Since join-irreducible elements generate all elements in a finite distributive lattice we have that
$\cl^{-1}\bigl(\big\uparrow\iota_\cX(\xi) \bigr) = \{ \cU^\topc\in \sJ(\sO(\sSC))\mid \xi\in \cl~\cU^\topc\}$ and thus

\[
\sJ(\cl)(\iota_\cX(\xi)) = \min\bigl\{ \cU^\topc\in \sJ(\sO(\sSC))\mid \xi\in \cl~\cU^\topc\bigr\},
\]

which is attained by a unique $\widehat \cU^\topc\in \sJ(\sO(\sSC))$.
Recall that $\cU^\topc\in \sJ(\sO(\sSC))$ if and only if $\cU^\topc=\big\downarrow\cS$, $\cS = [\xi^\topc]$ for some $\xi^\topc\in \cX^\topc$.
Therefore, $\xi\in \cl~\cU^\topc$ if and only if $\xi \in \cl~\eta^\topc$ for some $\eta^\topc\in \big\downarrow [\xi^\topc]$.
By duality $\xi \in \cl~\eta^\topc$ if and only if $\eta^\topc \in \st\xi\cap \cX^\topc$.
Consider the set

\[
\bigl\{[\eta^\topc]\in \sSC\mid \eta^\topc \in \st\xi\cap \cX^\topc \bigr\}.