Timeless state of gravity:
Black hole universal clock

Ahmed Farag Ali△†111Email<EMAIL_ADDRESS>
<EMAIL_ADDRESS>

†GRAVITON, San Mateo, CA 94401, USA

△Dept. of Physics, Benha University, Benha 13518, Egypt

Abstract

We investigate Rindler’s frame measurements. From its perspective, we found a
geometric/gravitational interpretation of speed of light, mass and uncertainty
principle. This can be interpreted as measurements of a black hole universal
clock. This lead to an emergence of a timeless state of gravity in a
mathematically consistent way. In other words, space my be a frozen time.

##  1 Clock Postulate and Global Reference Frame

One of the widely known incompatibilities between special relativity and
general relativity is how each theory sees red-shift. In special relativity,
the moving observers notice clocks rate change by a Lorentz factor $\gamma$.
But stationary observers share the same clocks rate even if they are at
different spatial positions as the electromagnetic wave frequency is only a
source-dependent property. Therefore, if those stationary observers notice any
shift in the wavelength, then it is justified only if the observer’s clock
ticks with different rates at different spacetime points, which is not
compatible of special relativity. In fact the general theory of relativity
justifies the change in clock rate as a response to the change in the
gravitational potential from point to another, i.e., gravitational red-shift
is a property of general covariance, which is global principle of general
relativity. That is why A. Einstein realized that special theory of relativity
and the equivalence principle hold locally, not globally [1]. This is clearly
manifested in accelerated Rindler spacetime with metric
$ds^{2}=-r^{2}dt^{2}+dr^{2}$ [8] such that the observer’s clock rate is
determined by $R(r)=\big{(}1-2V(r)\big{)}^{1/2}$, where $V(r)$ is the
gravitational potential. Therefore, when $r=0$, the observer clock rate is
also zero; $R(r)=0$, which matches with the expected observation of the clock
rate at the horizon: time freezes. As long as the gravitational field is
uniform and weak, we can approximately comparing
$1/\gamma=\sqrt{1-v^{2}/c^{2}}$ of special relativity to $R(r)\sim
1-\big{(}V(r)/c^{2}\big{)}$ in general relativity, where $g_{tt}=-R^{2}(r)$.
By a uniform field we mean it has one direction and constant value like the
electric field between two infinitely long parallel plates. And by weak we
mean $V(r)/c^{2}<<1$. This means locally we can explain the redshift assuming
special relativity and the equivalence principle. But when the full theory of
general relativity is assumed, things change. In special relativity you should
detect redshift while the source is moving with constant speed from spatial
point to another. This is Doppler shift and it’s constant because it’s
characterized by the speed only. However in the presence of gravity and
without source movement, we still can detect a gravitational redshift when the
time-varying gravitational potential itself changes at the same point through
time. The redshift is characterized by the change in the gravitational
potential. So if we want to study the change in the gravitational potential,
we need to consider some differential equation. If you want to solve this
differential equation from SR viewpoint, you don’t get correct answer. Rather
you get Abraham theory or Nordstrom theory. And if you do the differential
equation from GR viewpoint, i.e., general covariance differential equation,
you get the answer that matches with experiments. But that answer is not
compatible with SR as SR is based on inertial frames only. More details on
this issue is reported here [6].

In order to correlate how special relativity and general relativity see
redshift, we should ask how acceleration/gravitation affect the clock rate.
According to Don Koks [2], we need to take into consideration what he calls is
_clock postulate_. The clock postulate generalizes the special relativistic
comparison between the different clocks rates according to the Lorentz factor
$\gamma$. This generalization is based on that even if the moving clock
accelerates, the ratio of the rate of stationary clock compared to its rate is
still scaled or compared by $\gamma$ as in relativistic Doppler effect.
Consequently, ratio depends only on $v$ not the derivatives of $v$. Therefore,
an accelerating clock counts out its time in a way that fore every one moment,
its rate will slow by a factor $\gamma$ which is determined by instantaneous
velocity at that moment; its acceleration has no effect at all. It is
important to emphasize on that the clock postulate does not say that the
counting out rate of a moving clock is unaffected by its acceleration. It says
that the quantitative measurement of the relative rate does not depend on
acceleration. Equivalently, the rate of such accelerated clock is
indistinguishable from that of a clock in a _momentarily comoving inertial
frame_. For that, we can imagine an observer, in normal frame, is holding an
inertial clock that for a brief moment slows to a stop alongside another
accelerated observer with a clock, so that their relative velocity is
momentarily zero. This is crucial as we shall see when we consider The
Arnowitt-Deser-Misner (ADM) formalism. At that moment they are ticking at the
same rate. If that all is still confusing, you may consider the example of
riding a bicycle on an icy morning that is mentioned in Ref. [2].
Contemplation about the clock postulate is not that obvious as the postulate
can not be proved. Rather, simply it describes how we observe the physical
world. It might be thought that the clock postulate mandates abandoning the
Equivalence Principle as the Equivalence Principle equates between
gravitational fields and acceleration. However, this is not true for the
reasons explained in Ref. [2] by the example of a rocket empty of fuel and the
two astronauts.

##  2 One preferred frame

The absolute space and time means by definition a preferred frame, in which
physics laws appear to be recognizably different from they appear in other
frames. This Newtonian concept is contradicted with the assumed principle of
relativity in inertial frames and the principle of covariance in general
theory of relativity. However, all inertial frames are preferred over non-
inertial frames in special relativity as they observe only cause-and-effect
relations between events in closed intervals, a property that general
relativity lacks as it can observe a cause-and-effect relations between
internal and external events. In this paper, we use the Rindler observer and
the ADM formalism to restore the compatibility between the inertial privileged
frames and relative space of general relativity, at least slice by slice. For
more details on this issue, see [5, 6]. We consider spacetime foliation in ADM
formalism [7], at every given moment according to canonical observers, each
with its speed at that moment. These observers are related to each other by
“synchronicity” [3]. This approach comes with a global reference frame that
stays Minkowskian along the worldline of such observer. Despite that the
absolute synchronicity is not attainable either in special relativity or in
general relativity, a local synchronicity is extended beyond local
neighborhoods as in Ref. [3]. This extension shows how the observed redshift
can be useful in constructing a black hole universal clock with the help of
relative gravitational redshift, as how we show in this letter, and the idea
that the black hole is fully entangled with the space around it from Hawking
radiation. The entanglement “monogamy” is saved by the cool horizon approach
[4]. Like any clock, time will be measured in terms of lengths. This clock
will form a triangle that gives all measurements this clock can yield. The
Rindler observer [8] would analyze the ADM formalism as if the shift function
equal zero $N_{i}=0$. Therefore, the normal vector $\hat{n}$ to the spatial
slice is proportional to the time basis $\partial_{t}$ of Rindler frame field
with proportionality factor that is equal to the lapse function $\alpha$. In
this case the coordinate observer is moving with velocity
$(g_{tt})^{-1/2}e_{t}$ while the normal observer moves with velocity
$\hat{n}$. The factor similar to $\gamma$ in ADM formalism is in form of
$(1-N_{i}N^{i}/\alpha^{2})^{-1/2}$. See Fig 1. Rindler observer, the previous
factor is equal to 1. However this comparison between observers is NOT a
physical relativist Lorentz boost, and the frame of reference can be dragged
with any speed the observer chooses. It turns out that the normal observer and
the Rindler observer are _the same_ or _One Observer_. They both see each
other as if they are both belong to the same inertial Minkowski frame. This
matches with the global reference frame described in Ref. [3]. The one
observer idea may match with Wheeler idea of One-electron universe that he
described to Feynman and on which Feynman depend to assume that ”positrons
could simply be represented as electrons going from the future to the past in
a back section of their world lines” as Feynman stated in his Nobel speech
[9].

Figure 1: ADM slices of spacetime

##  3 The clock measurements

We assume an existence of Schwarzschild black hole with an event horizon. We
investigate the gravitational red-shift which is a property of general
covariance. The Rindler’s Observer measures the gravitational red-shift which
is a property of general covariance and its relative values between any two
points or slices of spacetime. Therefore, we consider two points ($A$ and $R$)
in the gravitational field of black hole as shown in the following Fig. (2).
Notice here these two points form a triangle that follow a geodesics geometry
of the considered black hole when connecting the two points with black hole
center. If $R$ and $A$ are far enough from $K$, the triangle become
approximately Euclidean triangle.

Figure 2: Black hole universal clock

Between these two points $A$ and $R$, there are two possible clock
measurements as follows:

  1. 1.

Relative gravitational red-shift which is represented by the ratio at two
different points

$\frac{z_{A}}{z_{R}}=\frac{(1-\frac{r_{s}}{r_{A}})^{-1/2}-1}{(1-\frac{r_{s}}{r_{R}})^{-1/2}-1}$
(1)

  2. 2.

The difference in gravitational red-shift at two different points.

$\Delta
z=z_{A}-z_{R}=(1-\frac{r_{s}}{r_{A}})^{-1/2}-(1-\frac{r_{s}}{r_{R}})^{-1/2}$
(2)

##  4 Weak Gravitational Approximation

###  4.1 Relative gravitational red-shift

We consider the weak gravitational approximations, $r_{s}<<r_{K}$ and
$r_{s}<<r_{R}$. The gravitational red-shift for both $A$ and $R$ can be
approximated as follows

$\displaystyle z_{A}$ $\displaystyle=$
$\displaystyle(1-\frac{r_{s}}{r_{A}})^{-1/2}-1\approx\frac{r_{s}}{2r_{A}}$ (3)
$\displaystyle z_{R}$ $\displaystyle=$
$\displaystyle(1-\frac{r_{s}}{r_{R}})^{-1/2}-1\approx\frac{r_{s}}{2r_{R}}.$
(4)

We compute the relative gravitational red-shift using Eq(1). We express it in
terms of all lengths measured at $R$ including the distance between $A$ and
$R$ ($r_{AR}$).

$\frac{z_{R}}{z_{A}}=\frac{1}{\sqrt{1-\frac{r_{AR}^{2}}{r_{A}^{2}}+2\frac{r_{R}r_{AR}}{r_{A}^{2}}\cos{\alpha}}}=\delta$
(5)

Notice the value of $\alpha$ can be $0\leq\alpha\leq\pi/2$. This equation
represents the relative gravitational-red-shift between two points $A$ and $R$
in a weak gravitational field. For the case $\alpha=\pi/2$. The relative
gravitational red-shift will be given by

$\frac{z_{R}}{z_{A}}=\frac{1}{\sqrt{1-(\frac{r_{AR}}{r_{A}})^{2}}}=\delta$ (6)

On the other side, the measurement in local inertial frames are determined in
terms of relative time dilation as follows

$\frac{t_{R}}{t_{A}}=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$ (7)

where $v$ is the relative speed between the two points $A$ and $R$ in the
local inertial frames and $c$ is the speed of light.

We notice the relative gravitational red-shift or relative gravitational time
dilation matches with the definition of time dilation in special relativity if
the ratio $r_{AR}^{2}/{r_{A}}^{2}$ can be replaced by ratio $v^{2}/c^{2}$. The
match is legitimate and mathematically consistent since the relative
gravitational red-shift introduces a local measurement which is the time
dilation in local inertial frames. Therefore, for mathematical consistency of
general relativity, its local measurement should be equivalent to measurement
special relativity that hold only in local inertial frames. This means when
$\alpha=\pi/2$, the gamma factor of special relativity emerges as a ratio
between the gravitational red-shift at $A$ and $R$. The clock measurements
depends only on “one variable”; the distance from the gravitational source,
which is the reason for velocity ratios turned to be lengths ratios in this
delta factor in Eq. (6). The ratio $r_{AR}^{2}/{r_{A}}^{2}$ can be considered
as a geometric or gravitational interpretation of the ratio $v^{2}/c^{2}$.
This comparison can be written as

$\frac{r_{AR}}{r_{A}}=\frac{r_{AR}/t}{r_{A}/t}=\frac{v}{c}$ (8)

This would support the approach of time varying speed of light as a solution
of cosmological puzzles that was suggested in [10]. It may support also the
experimental findings of changing physical constants such as fine structure
constant in gravitational field as shown recently in [11]. In our case, the
ratio $v/c$ varies depending on the distance from the gravitational source.

We note that time can be inserted easily in the previous equation as a
“redundant variable” which suggest a possible timeless state which is
consistent mathematically. The timeless state has been proposed in many
contexts such as shape dynamics which introduce a gravitational origin of
arrow of time [12, 13]. The timeless also emerged in Thermal time hypothesis
which assume that time only flow in thermodynamics or statistical patterns
[14]. It has been mathematically intuited as well that timeless universe is
possible [15]. In other words, space may be a frozen time [16].

To realize the effect of other values of angle $\alpha$ in weak gravitational
field, we consider an approximation which is $r_{AR}<<r_{A}$, $r_{AR}<<r_{R}$.
In that case, the delta factor in Eq. (5) is approximated as following

$\displaystyle\delta\approx 1-\frac{r_{R}r_{AR}}{r_{A}^{2}}\cos{\alpha}$ (9)

It is found that this equation matches with the derivative of Kepler equation.

$\frac{dm}{dE}=1-e\cos{E}$ (10)

where $m$ is the mean anomaly, $E$ is the eccentric anomaly, and $e$ is the
eccentricity. In our approximation, the eccentricity $e$ is approximately
equal to $r_{R}r_{AR}/r_{A}^{2}$, and $E$ refers to the angle $\alpha$. This
gives a geometric interpretation of Kepler equation from the relative
gravitational red-shift.

###  4.2 Difference in Gravitational red-shift

In this section, we compute the clock measurement as difference in
gravitational red-shift. For weak gravitational approximation, we get

$\Delta z=z_{R}-z_{A}=\frac{r_{s}}{2r_{R}}-\frac{r_{s}}{2r_{A}}$ (11)

Let us make an approximation as following $r_{A}=r_{R}+x$, where $x<<r_{A}$
and $x<<r_{R}$. In that case, Eq. (11) will be rewritten as follows. We use
the value of Schwarzschild radius $r_{s}=2GM/c^{2}$

$\Delta z=GM\frac{x}{c^{2}r_{R}^{2}}$ (12)

where $G$, is the gravitational constant, $M$, is the black hole mass and $c$
is the speed of light. From Eq.(8), $c$ can be set to equal to $r_{A}$ if we
take $t$ to be unity since we agree that t is a redundant factor through
matching local gravity measurement with local inertial frames. We find that
Eq. (12) can be arranged to take the following form

$\Delta z~{}M=\Delta M=GM^{2}\frac{x}{r_{A}^{2}r_{R}^{2}}$ (13)

where $\Delta M=\Delta z~{}M$. $\Delta M$ represents a possible mass between
any two different points in the gravitational field. This would give a
geometric representation for mass concept in terms of difference between
different points to the black hole. We want to understand the physical meaning
of the factor $GM^{2}$ in r.h.s of Eq. (13). When we look at Bekenstein-
Hawking entropy equation [17, 18].

$S_{BH}=\frac{c^{3}A}{4G\hbar}=\frac{4\pi}{c\hbar}GM^{2}$ (14)

where $A=16\pi(GM/c^{2})^{2}$ stands for surface area of a black hole. We
found that the factor $GM^{2}$ in r.h.s of Eq. (13) between any two different
points can be expressed in terms of black hole entropy as follows

$\Delta z~{}M=\Delta M=\frac{\hbar}{4\pi}\frac{x}{r_{R}^{2}r_{A}}S_{BH}$ (15)

We assumed that time is a unity. let us consider this unit as the Planck time.
This means that the Planck constant in previous equation can be replaced
through the following process

$t_{p}=\sqrt{\frac{\hbar G}{c^{5}}}=1$ (16)

Since the Planck time is our unity, then c can set to be $r_{A}$. Therefore,
the Planck constant in this geometric picture will be given by

$\hbar G=r_{A}^{5}$ (17)

This equation gives a geometric or gravitational interpretation of Planck
constant. The relative mass between any two different points is therefore
given by

$\Delta zM=\Delta M=\frac{1}{4\pi
G}\frac{x~{}r_{A}^{4}}{r_{R}^{2}}S_{BH}=\frac{1}{16\pi
G}\frac{x~{}~{}r_{A}^{2}}{r_{R}^{2}}~{}~{}A$ (18)

The previous equation gives purely a geometric expression for the relative
mass in terms of the gravitational source area of its full entropy. It is
experimentally proved that the difference in gravitational potential has an
effect on the apparent weight of the 14.4-keV ray of Iron (Fe) [19, 20]. This
may be an experimental support for the derived relation that connect the
difference in gravitational red-shift and emergence of mass in this section.

##  5 Gravity and Uncertainty

In previous sections, we have shown that the concept of velocity is replaced
with the relative distance between any any two different points in the space
time in the black hole universal clock. This would generate a timeless state
in a mathematically consistent way. In that state, the gravitational
measurements happens in terms of only one variable which is the distance from
the gravitational source. Time variable appear to be a redundant variable.
Since time, and therefore velocity dissolves, therefore there is no meaning to
define uncertainty in this timeless state. We propose that the distance from
the gravitational source may form the _hidden variable_ of every observation
process in quantum mechanics. This may complete the connection between quantum
mechanics and gravity in one unified theory, which is the timeless state. This
may complete the picture that was introduced in EPR [21]. This implies that
the uncertainty amount would decrease as the measurement happens closer to the
gravitational source. The uncertainty _emerges_ due to the difference in
information between point $A$ and point $R$ without knowing the distance to
the source. This difference is encoded in Eq. 18. The difference in
information (uncertainty) would be represented by the difference in
gravitational red-shift as follows:

$\frac{4\pi\Delta Mr_{R}^{2}r_{A}}{xS_{BH}}=\hbar$ (19)

Notice that the difference in information between Point $A$ and point $R$
depends on the distances $r_{A}$ and $r_{R}$. If we do not know these values,
this difference will be hidden in our local measurements, and therefore
uncertainty emerges. Notice that the variables on the left hand side are
greater than or equal to the Planck constant. This relation represent the
hidden variables which is reason for emergence of uncertainty principle
inequality in local measurements.

##  6 Strong Gravity Case

In strong gravity case, we can use Eqs. 1 and 2 without any approximation.
These relations can be computed for any two points, and it gives a wide
spectrum of measurements of relative gravitational red-shifts and masses in
strong gravity field. In strong gravity field, the triangle will not be
perfectly Euclidean but can be computed for every kind of measurement by
knowing the length of this triangle.

##  7 Why Clock Postulate/Timeless State of Gravity may be important?

Time in principle is the change happening for any physical system. To approach
a unified picture of physics, we can investigate states in which no change
happening, a frozen moment of the physical system. In that state, everything
will be reduced to geometry. In this geometric picture, we can find a
geometric picture of mass, speed of light, etc. Timeless state in that sense
could introduce a unified picture of different concepts in physics. We hope to
evolve this study in the future.

##  8 Conclusion

We investigate how Rindler observer in ADM formalism make measurements in the
slices of space-time. We found that the Rindler observer in ADM formalism
restores the compatibility between the inertial privileged frames and relative
space of general relativity, at least slice by slice. We determined the
relative or local measurements that Rindler observer could make in each slice
of the space-time. We got a timeless state of the universe in which we found a
geometric interpretations of speed of light and mass. The timeless state may
form a correlation between relative gravitational red-shift and internal
symmetries that are independent of time. We hope that this equivalence may
open the door for a gravitational technology. It is worth mentioning that
timeless state of gravity can shed a light on the nature of problem of time in
Wheeler-De-Witt equation. We hope to report on these in the future.

Acknowledgment This work is supported by the quantum gravity research grant,
Los Angeles, California.

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# Hofmann-Mislove theorem for approach spaces††thanks: This work is supported
by the National Natural Science Foundation of China (No. 11871358)

Junche Yu, Dexue Zhang
School of Mathematics, Sichuan University, Chengdu 610064, China
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The Hofmann-Mislove theorem says that compact saturated sets of a sober
topological space correspond bijectively to open filters of its open set
lattice. This paper concerns an analogy of this result for approach spaces. To
this end, the notion of compact functions of approach spaces is introduced.
Such functions are an analogue of compact subsets in the enriched context. It
is shown that for a sober approach space, the inhabited and saturated compact
functions correspond bijectively to the proper open $[0,\infty]$-filters of
the metric space of its upper regular functions, which is an analogy of the
Hofmann-Mislove theorem for approach spaces.

Keywords Approach space, compact function, sober approach space, metric space,
$[0,\infty]$-ideal, open $[0,\infty]$-filter

MSC(2020) 18B35, 18F60, 54A05, 54B30

##  1 Introduction

While metric spaces (not necessarily symmetric and finitary) are ordered sets
(or, categories) enriched over Lawvere’s quantale $([0,\infty],\geq,+,0)$
[18], approach spaces introduced by Lowen [22] are topological spaces enriched
over $([0,\infty],\geq,+,0)$. Approach spaces are among the prime examples in
_Monoidal Topology_ (see e.g. [5, 11, 12, 13]). For an exposition of these
spaces, we refer to the monograph [24] by their inventor.

This paper concerns the connections between metric spaces and approach spaces,
such connections are an important ingredient of Quantitative Domain Theory
which, roughly speaking, replaces partially ordered sets in Domain Theory [25,
8] by metric spaces and more generally, quasi-uniform spaces (see e.g. [3, 9,
26, 30]). It has become clear that a coherent theory, parallel to that of
domains, in the $[0,\infty]$-enriched context is needed; see [1, 10, 14, 20,
31] for some efforts in this regard. Of course, necessary modifications have
to be made in the $[0,\infty]$-enriched context; these modifications often
help us understand the subject better.

An important result in domain theory, known as the Hofmann-Mislove theorem
[15, 8], says that the (non-empty) compact saturated sets of a sober
topological space $X$ correspond bijectively to the (proper) open filters of
the complete lattice $\mathcal{O}(X)$ of open subsets of $X$, hence
establishes a connection between objects of a topological nature to objects of
an order-theoretic nature. The aim of this paper is to look for an analogy of
this theorem in the $[0,\infty]$-enriched context. In order to achieve this,
we have to determine, in the $[0,\infty]$-enriched context, an analogue of
compact subset and an analogue of open filter of open-set lattice.

In [1] a notion of open $[0,\infty]$-filter of metric spaces is introduced.
There is a natural metric $\rho_{X}$ on the set $\mathfrak{U}X$ of upper
regular functions of an approach space, so, we take open $[0,\infty]$-filters
of the metric space $(\mathfrak{U}X,\rho_{X})$ for $[0,\infty]$-enriched
version of open filters of open-set lattices. It remains to determine an
analogue of compact subset in the enriched context. To this end, the notion of
compact functions of approach spaces is introduced (Definition 3.9) and some
basic properties of such functions are investigated. The results show that
such functions behave in a way parallel to that of compact subsets of
topological spaces.

compact subset | lattice of open sets | open filter
---|---|---
compact function | metric space of upper regular functions | open $[0,\infty]$-filter

With the notions of compact functions and open $[0,\infty]$-filters at hand,
we establish an enriched version of the Hofmann-Mislove theorem in the last
section, which says that for a sober approach space $(X,\delta)$, the proper
open $[0,\infty]$-filters of the metric space $(\mathfrak{U}X,\rho_{X})$ of
upper regular functions correspond bijectively to the inhabited compact
saturated functions of $(X,\delta)$.

##  2 $[0,\infty]$-ideals and $[0,\infty]$-filters of metric spaces

In this section we recall some basic notions about metric spaces, viewed as
categories enriched over Lawvere’s quantale $([0,\infty],\geq,+,0)$ [18, 4],
the aim is to fix terminologies and notations. Only Lemma 2.6, Proposition
2.10 and Proposition 2.11 are new.

A metric space is a pair $(X,d)$ consisting of a set $X$ and a map $d\colon
X\times X\longrightarrow[0,\infty]$, called a metric on $X$, such that
$d(x,x)=0$ and $d(x,y)+d(y,z)\geq d(x,z)$ for all $x,y,z\in X$. For each
metric $d$ on $X$, $d^{\rm op}(x,y)\coloneqq d(y,x)$ is also a metric on $X$,
called the opposite of $d$. A metric $d$ is

  * •

symmetric, if $d=d^{\rm op}$; that is, $d(x,y)=d(y,x)$ for all $x,y\in X$;

  * •

separated, if $x=y$ whenever $d(x,y)=d(y,x)=0$;

  * •

finitary, if $d(x,y)<\infty$ for all $x,y\in X$.

A “classical metric” is just a symmetric, separated and finitary one.

A map $f\colon(X,d_{X})\longrightarrow(Y,d_{Y})$ between metric spaces is
_non-expansive_ if for all $x,y\in X$,

$d_{X}(x,y)\geq d_{Y}(f(x),f(y)).$

A map $f\colon(X,d_{X})\longrightarrow(Y,d_{Y})$ between metric spaces is
_isometric_ if for all $x,y\in X$,

$d_{X}(x,y)=d_{Y}(f(x),f(y)).$

Non-expansive maps are just $[0,\infty]$-functors when metric spaces are
viewed as $[0,\infty]$-categories. Metric spaces and non-expansive maps form a
category

$\sf Met.$

The _underlying order_ $\sqsubseteq_{d}$ of a metric space $(X,d)$ is the
order relation on $X$ defined by

$x\sqsubseteq_{d}y\quad\text{if $d(x,y)=0.$}$

It is clear that a metric space is separated if and only if its underlying
order is a partial order and that any non-expansive map preserves the
underlying order.

For all $a,b\in[0,\infty]$, let

$d_{L}(a,b)=b\ominus a\coloneqq\max\\{0,b-a\\}.$

Then $d_{L}$ is a metric on $[0,\infty]$. The opposite of $d_{L}$ is denoted
by $d_{R}$; that is,

$d_{R}(a,b)=a\ominus b.$

Both ${([0,\infty],d_{L})}$ an ${([0,\infty],d_{R})}$ are separated, non-
symmetric, and non-finitary metric spaces. In general, for each set $X$, the
map

$\rho_{X}\colon[0,\infty]^{X}\times[0,\infty]^{X}\longrightarrow[0,\infty],\quad\rho_{X}(\lambda,\mu)=\sup_{x\in
X}\mu(x)\ominus\lambda(x)$

is a metric on the set $[0,\infty]^{X}$. If $X$ is a singleton set then the
metric space $([0,\infty]^{X},\rho_{X})$ degenerates to $([0,\infty],d_{L})$.

The underlying order $\sqsubseteq$ of $([0,\infty]^{X},\rho_{X})$ is given by

$\lambda\sqsubseteq\mu\iff\forall x\in X,\lambda(x)\geq\mu(x).$

In particular, the underlying order of $([0,\infty],d_{L})$ is opposite to the
usual less-than-or-equal-to relation.

######  Convention.

Some conventions and notations.

  1. (a)

When talking about a subset $A$ of $[0,\infty]^{X}$, we always assume that $A$
is endowed with the metric inherited from the space
$([0,\infty]^{X},\rho_{X})$, which will also be denoted by $\rho_{X}$.

  2. (b)

Though the underlying order of $([0,\infty]^{X},\rho_{X})$ is opposite to the
_usual_ pointwise order, for each subset $\\{\lambda_{i}\\}_{i}$ of
$[0,\infty]^{X}$, we still write $\inf_{i}\lambda_{i}$ and
$\sup_{i}\lambda_{i}$, respectively, for its greatest lower bound and least
upper bound with respect to the usual pointwise order.

  3. (c)

For each $r\in[0,\infty]$ and $A\subseteq X$, we write $r_{A}$ for the map
$X\longrightarrow[0,\infty]$ given by

$r_{A}(x)=\begin{cases}r,&x\in A,\\\ \infty,&x\notin A.\end{cases}$

Assume that $(X,d_{X})$, $(Y,d_{Y})$ are metric spaces; $f\colon
X\longrightarrow Y$ and $g\colon Y\longrightarrow X$ are non-expansive maps.
We say that $f$ is left adjoint to $g$, or $g$ is right adjoint to $f$, and
write $f\dashv g$, if for all $x\in X$ and $y\in Y$,

$d_{Y}(f(x),y)=d_{X}(x,g(y)).$

Let $(X,d)$ be a metric space. A weight of $(X,d)$ is a map $\phi\colon
X\longrightarrow[0,\infty]$ such that for all $x,y\in X$,

$\phi(y)+d(x,y)\geq\phi(x).$

Said differently, a weight of $X$ is a non-expansive map
$\phi\colon(X,d)\longrightarrow{([0,\infty],d_{R})}$. We write

$\mathcal{P}X$

for the set of all weights of $(X,d)$.

A _coweight_ of $(X,d)$ is defined to be a weight of $(X,d^{\rm op})$.
Explicitly, a coweight of $(X,d)$ is a map $\lambda\colon
X\longrightarrow[0,\infty]$ such that for all $x,y\in X$,

$\lambda(x)+d(x,y)\geq\lambda(y).$

For each element $x$ of a metric space $(X,d)$, the map

$d(-,x)\colon X\longrightarrow[0,\infty]$

is a weight of $(X,d)$. Weights of this form are said to be _representable_.

######  Lemma 2.1 (Yoneda lemma).

Let $(X,d)$ be a metric space. Then for each point $x$ and each weight $\phi$
of $(X,d)$,

$\rho_{X}(d(-,x),\phi)=\phi(x).$

The map

${{\sf y}}\colon(X,d)\longrightarrow(\mathcal{P}X,\rho_{X}),\quad{\sf
y}(x)=d(-,x)$

is an isometry by the Yoneda lemma, and is called the _Yoneda embedding_.

Each non-expansive map $f\colon X\longrightarrow Y$ gives rise to an
adjunction between $\mathcal{P}X$ and $\mathcal{P}Y$. Precisely, the map

$f^{\rightarrow}\colon\mathcal{P}X\longrightarrow\mathcal{P}Y,\quad
f^{\rightarrow}(\phi)(y)=\inf_{x\in X}(\phi(x)+d_{Y}(y,f(x))$

is left adjoint to

$f^{\leftarrow}\colon\mathcal{P}Y\longrightarrow\mathcal{P}X,\quad
f^{\leftarrow}(\psi)(x)=\psi(f(x)).$

Let $(X,d)$ be a metric space and let $\phi$ be a weight of $(X,d)$. An
element $a$ of $X$ is called a _colimit_ of $\phi$ if

$d(a,y)=\rho_{X}(\phi,d(-,y))$

for all $y\in X$. Any colimits $a$ and $b$ of the same weight are isomorphic
in the sense that $d(a,b)=d(b,a)=0$. Dually, an element $a$ of $X$ is called a
_limit_ of a coweight $\psi$ if

$d(y,a)=\rho_{X}(\psi,d(y,-))$

for all $y\in X$.

A metric space $(X,d)$ is _cocomplete_ if every weight of $(X,d)$ has a
colimit. One readily checks that $(X,d)$ is cocomplete if and only if the
Yoneda embedding ${\sf y}\colon(X,d)\longrightarrow(\mathcal{P}X,\rho_{X})$
has a left adjoint. It is known (see e.g. [27]) that $(X,d)$ is cocomplete if
and only if $(X,d)$ is _complete_ in the sense that every coweight has a
limit.

Let $(X,d)$ be a metric space. For each $x\in X$ and $r\in[0,\infty]$, the
_tensor of $r$ with $x$_, denoted by $r\otimes x$, is a point of $X$ such that
for all $y\in X$,

$d(r\otimes x,y)=d(x,y)\ominus r.$

The _cotensor of $r$ with $x$_, is an element $r\rightarrowtail x$ of $X$ such
that for all $y\in X$,

$d(y,r\rightarrowtail x)=d(y,x)\ominus r.$

A metric space $(X,d)$ is _tensored_ if the tensor $r\otimes x$ exists for all
$x\in X$ and $r\in[0,\infty]$. Every cocomplete metric space is tensored:
$r\otimes x$ is the colimit of the weight $r+d(-,x)$.

The following characterization of cocomplete (hence complete) metric spaces is
a special case of [28, Theorem 2.7], the conditions (ii) and (iii) amount to
that $(X,d)$, as an enriched category, is conically cocomplete.

######  Proposition 2.2.

A metric space $(X,d)$ is cocomplete if and only if

  1. (i)

$(X,d)$ is tensored;

  2. (ii)

each subset $A$ of $X$ has a join in $(X,\sqsubseteq_{d})$; and

  3. (iii)

for each $x\in X$ and each $A\subseteq X$, $d(\bigvee A,x)=\sup\limits_{a\in
A}d(a,x)$, where $\bigvee A$ refers to the join of $A$ in
$(X,\sqsubseteq_{d})$.

######  Example 2.3.

For each set $X$, the metric space $([0,\infty]^{X},\rho_{X})$ is cocomplete,
hence complete. In particular, for all $\lambda\in[0,\infty]^{X}$ and
$r\in[0,\infty]$, $r\otimes\lambda=r+\lambda$ and
$r\rightarrowtail\lambda=\lambda\ominus r$.

######  Proposition 2.4.

If $(X,d_{X})$ and $(Y,d_{Y})$ are both tensored metric spaces, then a map
$f\colon(X,d_{X})\longrightarrow(Y,d_{Y})$ is non-expansive if and only if

  1. (i)

$f\colon(X,\sqsubseteq_{d_{X}})\longrightarrow(Y,\sqsubseteq_{d_{Y}})$
preserves order; and

  2. (ii)

for all $x\in X$ and $r\in[0,\infty]$, $r\otimes
f(x)\sqsubseteq_{d_{Y}}f(r\otimes x)$.

###### Proof.

Routine.∎

For each set $X$, define a function

$\operatorname{\pitchfork}\colon[0,\infty]^{X}\times[0,\infty]^{X}\longrightarrow[0,\infty]$

by

$\phi\operatorname{\pitchfork}\lambda=\inf_{x\in X}(\phi(x)+\lambda(x))$

for all $\phi,\lambda\in[0,\infty]^{X}$. This function is closely related to
the metric $\rho_{X}$ on $[0,\infty]^{X}$ and plays an important role in this
paper.

######  Remark 2.5.

Let $(X,d)$ be a metric space; let $\phi$ be a weight and $\lambda$ be a
coweight of $(X,d)$. Then, in the terminology of category theory, the value
$\phi\operatorname{\pitchfork}\lambda$ is the $\phi$-weighted colimit of the
$[0,\infty]$-functor $\lambda\colon(X,d)\longrightarrow([0,\infty],d_{L})$;
that is,
$\phi\operatorname{\pitchfork}\lambda=\operatorname*{colim}\lambda^{\rightarrow}(\phi)$.
Furthermore, if we view $\phi$ as a distributor from $(X,d)$ to a singleton
metric space and view $\lambda$ as a distributor from a singleton metric space
to $(X,d)$ (see. e.g. [27]), then $\phi\pitchfork\lambda$ is essentially the
composite distributor $\phi\circ\lambda$.

######  Lemma 2.6.

Let $X$ be a set and $\phi,\lambda\in[0,\infty]^{X}$.

  1. (i)

$\phi\operatorname{\pitchfork}\lambda=\sup\limits_{r\in[0,\infty)}(\inf\\{r_{X},\phi\\}\operatorname{\pitchfork}\lambda).$

  2. (ii)

If $\lambda$ is bounded, then
$\rho_{X}(\phi,b\ominus\lambda)=b\ominus(\lambda\operatorname{\pitchfork}\phi)$
for each $b<\infty$ with $\lambda\leq b_{X}$.

  3. (iii)

If $\phi\operatorname{\pitchfork}\lambda<\infty$, then
$\phi\operatorname{\pitchfork}\lambda=b\ominus\rho_{X}(\phi,b\ominus\lambda)$
for each $b<\infty$ with $\phi\operatorname{\pitchfork}\lambda\leq b$.

###### Proof.

Routine calculations. ∎

Let $\lambda\in[0,\infty]^{X}$. We say that (i) $\lambda$ is _inhabited_ if
$\lambda\operatorname{\pitchfork}0_{X}=0$; and (ii) $\lambda$ is _vacuous_ if
$\lambda\operatorname{\pitchfork}\mu=\infty$ for all $\mu\in[0,\infty]^{X}$.
It is clear that $\lambda$ is inhabited if and only if $\inf\limits_{x\in
X}\lambda(x)=0$ and that $\lambda$ is vacuous if and only if
$\lambda=\infty_{X}$.

Let $(X,d)$ be a metric space. A net $\\{x_{i}\\}_{i}$ in $(X,d)$ is _forward
Cauchy_ [26, 3] if

$\inf_{i}\sup_{k\geq j\geq i}d(x_{j},x_{k})=0.$

######  Definition 2.7.

([6, 1]) A $[0,\infty]$-ideal of a metric space $(X,d)$ is a weight $\phi$
such that

$\phi=\inf_{i}\sup_{j\geq i}d(-,x_{j})$

for some forward Cauchy net $\\{x_{i}\\}_{i}$.

The following theorem characterizes $[0,\infty]$-ideals of a metric space
$(X,d)$ via their relation to the weights and coweights of $(X,d)$.

######  Theorem 2.8.

Let $\phi$ be a weight of a metric space $(X,d)$. The following are
equivalent:

  1. (1)

$\phi$ is a $[0,\infty]$-ideal.

  2. (2)

$\phi$ is inhabited and is irreducible in the sense that for all weights
$\phi_{1},\phi_{2}$ of $(X,d)$,

$\inf\\{\phi_{1},\phi_{2}\\}\leq\phi\implies\text{ either $\phi_{1}\leq\phi$
or $\phi_{2}\leq\phi$}.$

  3. (3)

$\phi$ is inhabited and is flat in the sense that for all coweights
$\lambda_{1},\lambda_{2}$ of $(X,d)$,

$\phi\pitchfork\sup\\{\lambda_{1},\lambda_{2}\\}=\max\\{\phi\pitchfork\lambda_{1},\phi\pitchfork\lambda_{2}\\}.$

###### Proof.

The equivalence $(1)\Leftrightarrow(3)$ is contained in the proof of [29,
Theorem 7.15]; the equivalence $(1)\Leftrightarrow(2)$ is a special case of
[17, Theorem 3.13]. ∎

Since each ideal $D$ of $(X,\sqsubseteq_{d})$ can be viewed as a forward
Cauchy net of $(X,d)$, then

$d(-,D)\coloneqq\inf_{x\in D}d(-,x)$

is a $[0,\infty]$-ideal of $(X,\delta)$. For a cocomplete metric space, every
$[0,\infty]$-ideal is of this form.

######  Proposition 2.9.

([16, Proposition 4.8]) Let $(X,d)$ be a cocomplete metric space and let
$\phi$ be a $[0,\infty]$-ideal of $(X,d)$. Then

  1. (i)

$\Delta(\phi)\coloneqq\\{x\in X\mid\phi(x)=0\\}$ is an ideal in
$(X,\sqsubseteq_{d})$ and $\phi=\inf\limits_{x\in\Delta(\phi)}d(-,x)$.

  2. (ii)

Colimits of $\phi$ in $(X,d)$ are precisely joins of $\Delta(\phi)$ in
$(X,\sqsubseteq_{d})$.

The following characterization of $[0,\infty]$-ideals of a cocomplete metric
space and its dual (Proposition 2.11) are useful in this paper.

######  Proposition 2.10.

Let $(X,d)$ be a cocomplete metric space. Then a function $\phi\colon
X\longrightarrow[0,\infty]$ is a $[0,\infty]$-ideal if and only if it
satisfies:

  1. (i)

for all $x\in X$ and $r\in[0,\infty]$, $\phi(r\otimes x)=\phi(x)\ominus r$;

  2. (ii)

$\phi(x\vee y)=\max\\{\phi(x),\phi(y)\\}$ for all $x,y\in X$, where $x\vee y$
is a join of $x$ and $y$ in $(X,\sqsubseteq_{d})$.

###### Proof.

If $\phi$ is a $[0,\infty]$-ideal, there is a forward Cauchy net
$\\{x_{i}\\}_{i\in I}$ of $(X,d)$ such that

$\phi=\inf_{i}\sup_{j\geq i}d(-,x_{j}).$

Now we check that it satisfies (i) and (ii).

(i) We calculate:

$\displaystyle\phi(r\otimes x)$ $\displaystyle=\inf_{i}\sup_{j\geq
i}d(r\otimes x,x_{j})$ $\displaystyle=\inf_{i}\sup_{j\geq i}(d(x,x_{j})\ominus
r)$ $\displaystyle=(\inf_{i}\sup_{j\geq i}d(x,x_{j}))\ominus r$
$\displaystyle=\phi(x)\ominus r.$

(ii) We calculate:

$\displaystyle\phi(x\vee y)$ $\displaystyle=\inf_{i}\sup_{j\geq i}d(x\vee
y,x_{j})$ $\displaystyle=\inf_{i}\sup_{j\geq
i}\max\\{d(x,x_{j}),d(y,x_{j})\\}$ (Proposition 2.2)
$\displaystyle=\inf_{i}\max\\{\sup_{j\geq i}d(x,x_{j}),\sup_{j\geq
i}d(y,x_{j})\\}$ $\displaystyle=\max\\{\inf_{i}\sup_{j\geq
i}d(x,x_{j}),\inf_{i}\sup_{j\geq i}d(y,x_{j})\\}$ (the index set $I$ is
directed) $\displaystyle=\max\\{\phi(x),\phi(y)\\}.$

To see the sufficiency, assume that $\phi\colon X\longrightarrow[0,\infty]$
satisfies (i) and (ii). First, by (i), (ii) and Proposition 2.4 one sees that
$\phi\colon(X,d)\longrightarrow([0,\infty],d_{R})$ is non-expansive, hence a
weight of $(X,d)$. Next, let

$\Delta(\phi)=\\{x\in X\mid\phi(x)=0\\}.$

By (ii) one sees that $\Delta(\phi)$ is an ideal of $(X,\sqsubseteq_{d})$. Now
we show that for each $x\in X$,

$\phi(x)=\inf_{a\in\Delta(\phi)}d(x,a),$

which implies that $\phi$ is a $[0,\infty]$-ideal since $\Delta(\phi)$ can be
viewed as a forward Cauchy net of $(X,d)$. On one hand, for each
$a\in\Delta(\phi)$, since $\phi(x)\leq\phi(a)+d(x,a)=d(x,a)$, it follows that

$\phi(x)\leq\inf_{a\in\Delta(\phi)}d(x,a).$

On the other hand, since $\phi(\phi(x)\otimes x)=\phi(x)\ominus\phi(x)=0,$
then $\phi(x)\otimes x\in\Delta(\phi)$. Since

$0=d(\phi(x)\otimes x,\phi(x)\otimes x)=d(x,\phi(x)\otimes x)\ominus\phi(x),$

it follows that $\phi(x)\geq d(x,\phi(x)\otimes x)$, hence

$\displaystyle\phi(x)$ $\displaystyle\geq\inf_{a\in\Delta(\phi)}d(x,a).\qed$

Following [1], a map $\lambda\colon X\longrightarrow[0,\infty]$ is called a
_$[0,\infty]$ -filter_ of $(X,d)$ if it is a $[0,\infty]$-ideal of $(X,d^{\rm
op})$. In other words, a $[0,\infty]$-filter of $(X,d)$ is a coweight
$\lambda$ such that

$\lambda=\inf_{i}\sup_{j\geq i}d(x_{j},-)$

for some net $\\{x_{i}\\}_{i\in I}$ that is _backward Cauchy_ in the sense
that

$\inf_{i}\sup_{k\geq j\geq i}d(x_{k},x_{j})=0.$

The following proposition is dual to Proposition 2.10.

######  Proposition 2.11.

Let $(X,d)$ be a complete metric space. Then a function $\lambda\colon
X\longrightarrow[0,\infty]$ is a $[0,\infty]$-filter if and only if it
satisfies:

  1. (i)

for all $x\in X$ and $r\in[0,\infty]$, $\lambda(r\rightarrowtail
x)=\lambda(x)\ominus r$;

  2. (ii)

$\lambda(x\land y)=\max\\{\lambda(x),\lambda(y)\\}$ for all $x,y\in X$, where
$x\land y$ is a meet of $x$ and $y$ in $(X,\sqsubseteq_{d})$.

Let $f\colon(X,d_{X})\longrightarrow(Y,d_{Y})$ be a non-expansive map. It is
seen readily that if $\phi$ is a $[0,\infty]$-ideal of $(X,d_{X})$, then
$f^{\rightarrow}(\phi)$ is a $[0,\infty]$-ideal of $(Y,d_{Y})$. We say that
$f$ is _$[0,\infty]$ -Scott continuous_ if $f$ preserves colimits of
$[0,\infty]$-ideals [1]; that is, for each $[0,\infty]$-ideal $\phi$ of
$(X,d_{X})$,
$f(\operatorname*{colim}\phi)=\operatorname*{colim}f^{\rightarrow}(\phi)$
whenever $\operatorname*{colim}\phi$ exists.

In the literature, $[0,\infty]$-Scott continuous maps are also known as
_Yoneda continuous_ maps. Here is the reason. An element $x$ of a metric space
$(X,d)$ is called a _Yoneda limit_ of a forward Cauchy net $\\{x_{i}\\}_{i}$
if for all $y\in X$,

$d(x,y)=\inf_{i}\sup_{j\geq i}d(x_{j},y).$

It is known (see e.g. [7]) that $x$ is a Yoneda limit of a forward Cauchy net
$\\{x_{i}\\}_{i}$ if and only if $x$ is a colimit of the $[0,\infty]$-ideal
$\inf_{i}\sup_{j\geq i}d(-,x_{j})$. So, a non-expansive map preserves Yoneda
limits (hence Yoneda continuous) if and only if it is $[0,\infty]$-Scott
continuous.

From Proposition 2.9 one immediately derives the following:

######  Proposition 2.12.

([16]) Let $(X,d_{X})$ and $(Y,d_{Y})$ be cocomplete metric spaces. Then a
non-expansive map $f\colon(X,d_{X})\longrightarrow(Y,d_{Y})$ is
$[0,\infty]$-Scott continuous if and only if, as an order-preserving map from
$(X,\sqsubseteq_{d_{X}})$ to $(Y,\sqsubseteq_{d_{Y}})$, $f$ is Scott
continuous.

##  3 Compact functions of approach spaces

While metric spaces are $[0,\infty]$-valued ordered sets, approach spaces are
$[0,\infty]$-valued topological spaces.

######  Definition 3.1.

([22, 24]) An approach space is a set $X$ together with a map $\delta\colon
X\times 2^{X}\longrightarrow[0,\infty]$, subject to the following conditions:
for all $x\in X$ and $A,B\in 2^{X}$,

  1. (A1)

$\delta(x,\\{x\\})$=0;

  2. (A2)

$\delta(x,\varnothing)=\infty$;

  3. (A3)

$\delta(x,A\cup B)=\min\\{\delta(x,A),\delta(x,B)\\}$;

  4. (A4)

$\delta(x,A)\leq\sup\limits_{b\in B}\delta(b,A)+\delta(x,B)$.

The value $\delta(x,A)$, called the distance from $x$ to $A$, measures how far
$x$ is to $A$.

A contraction (or, a continuous map)
$f\colon(X,\delta_{X})\longrightarrow(Y,\delta_{Y})$ between approach spaces
is a map $f\colon X\longrightarrow Y$ such that

$\delta_{X}(x,A)\geq\delta_{Y}(f(x),f(A))$

for all $A\subseteq X$ and $x\in X$.

Approach spaces and contractions form a category

$\sf App.$

######  Example 3.2.

([23, 24]) For all $x\in[0,\infty]$ and $A\subseteq[0,\infty]$, let

$\delta_{\mathbb{P}}(x,A)=\begin{cases}x\ominus\sup A,&A\neq\varnothing,\\\
\infty,&A=\varnothing.\end{cases}$

Then $\delta_{\mathbb{P}}$ is a distance on $[0,\infty]$. The space
$([0,\infty],\delta_{\mathbb{P}})$ is denoted by $\mathbb{P}$.

Like topological spaces, approach spaces can be described in many equivalent
ways. Among them we need the characterizations by _lower regular functions_
and _upper regular functions_. While the distance function of an approach
space $(X,\delta)$ is a $[0,\infty]$-valued analogy of the closure operator of
a topological space, lower regular functions and upper regular functions are
respectively analogies of closed sets and open sets in the
$[0,\infty]$-enriched context.

topological space | closure operator | open set | closed set
---|---|---|---
approach space | distance function | upper regular function | lower regular function

######  Definition 3.3.

[23, 24] A collection $\mathfrak{L}\subseteq[0,\infty]^{X}$ is called a lower
regular function frame if it satisfies:

  1. (L1)

For each subset $\\{\phi_{i}\\}_{i\in I}$ of $\mathfrak{L}$, $\sup_{i\in
I}\phi_{i}\in\mathfrak{L}$.

  2. (L2)

For all $\phi,\psi\in\mathfrak{L}$, $\inf\\{\phi,\psi\\}\in\mathfrak{L}X$.

  3. (L3)

For all $\phi\in\mathfrak{L}$ and $r\in[0,\infty]$, both $\phi+r$ and
$\phi\ominus r$ are in $\mathfrak{L}$.

For each approach space $(X,\delta)$, the set $\mathfrak{L}X$ of all
contractions $(X,\delta)\longrightarrow\mathbb{P}$ is a lower regular function
frame, where $\mathbb{P}$ is the approach space given in Example 3.2. Elements
of $\mathfrak{L}X$ are called _lower regular functions_ of $(X,\delta)$.
Conversely, if $\mathfrak{L}\subseteq[0,\infty]^{X}$ is a lower regular
function frame, then $X$ together with the map

$\delta\colon X\times
2^{X}\longrightarrow[0,\infty],\quad\delta(x,A)=\sup\\{\phi(x)\mid\phi\in\mathfrak{L},\forall
a\in A,\phi(a)=0\\}$

is an approach space with $\mathfrak{L}$ being its set of lower regular
functions. Thus, every approach space is determined by its lower regular
functions (see e.g. [23, 24]).

Let $(X,\delta)$ be an approach space and $x\in X$. It is readily verified
that $\delta(-,\\{x\\})$ is a lower regular function and

$\rho_{X}(\delta(-,\\{x\\}),\phi)=\phi(x)$ (3.i)

for each lower regular function $\phi$ of $(X,\delta)$.

For each approach space $(X,\delta)$, let

$\mathfrak{U}X=\\{\infty_{X}\\}\cup\Big{\\{}\lambda\in[0,\infty]^{X}\mid\sup_{x\in
X}\lambda(x)<\infty~{}\text{and}~{}r\ominus\lambda\in\mathfrak{L}X~{}\text{for
all}~{}r\in[0,\infty]\Big{\\}}.$

It is not hard to check that

$\mathfrak{U}X=\\{\infty_{X}\\}\cup\Big{\\{}\lambda\in[0,\infty]^{X}\mid\sup_{x\in
X}\lambda(x)<\infty~{}\text{and}~{}\Big{(}\sup_{x\in
X}\lambda(x)\Big{)}\ominus\lambda\in\mathfrak{L}X\Big{\\}}$

and that $\mathfrak{U}X$ satisfies the following conditions:

  1. (U1)

$\infty_{X}\in\mathfrak{U}X$; and every non-vacuous element (i.e., not equal
to $\infty_{X}$) of $\mathfrak{U}X$ is bounded.

  2. (U2)

For each subset $\\{\lambda_{i}\\}_{i\in I}$ of $\mathfrak{U}X$, $\inf_{i\in
I}\lambda_{i}\in\mathfrak{U}X$.

  3. (U3)

For all $\lambda,\mu\in\mathfrak{U}X$,
$\sup\\{\lambda,\mu\\}\in\mathfrak{U}X$.

  4. (U4)

For all $\lambda\in\mathfrak{U}X$ and $r\in[0,\infty]$, both $\lambda+r$ and
$\lambda\ominus r$ are in $\mathfrak{U}X$.

Elements of $\mathfrak{U}X$ are called _upper regular functions_ of
$(X,\delta)$ [24]. We hasten to note that the definition given here is
slightly different from that in [24], the difference is that the function
$\infty_{X}$, which is an analogy of the empty set, is postulated to be upper
regular. It is known (see e.g. [24]) that $\mathfrak{L}X$ is the smallest
lower regular function frame that contains

$\\{r\ominus\lambda\mid r\in[0,\infty],\lambda\in\mathfrak{U}X\\}.$

Thus, every approach space is also characterized by its upper regular
functions.

######  Proposition 3.4.

([24, Theorem 1.3.3]) For a map
$f\colon(X,\delta_{X})\longrightarrow(Y,\delta_{Y})$ between approach spaces,
the following are equivalent:

  1. (1)

$f$ is a contraction.

  2. (2)

For each lower regular function $\phi$ of $(Y,\delta_{Y})$, $\phi\circ f$ is a
lower regular function of $(X,\delta_{X})$.

  3. (3)

For each upper regular function $\lambda$ of $(Y,\delta_{Y})$, $\lambda\circ
f$ is an upper regular function of $(X,\delta_{X})$.

######  Proposition 3.5.

For each approach space $(X,\delta)$, both $(\mathfrak{L}X,\rho_{X})$ and
$(\mathfrak{U}X,\rho_{X})$ are cocomplete metric spaces.

###### Proof.

Actually, $\mathfrak{L}X$ is closed in $([0,\infty]^{X},\rho_{X})$ under the
formation of limits and $\mathfrak{U}X$ is closed in
$([0,\infty]^{X},\rho_{X})$ under the formation of colimits. In particular,
for each $\phi\in\mathfrak{L}X$ and $r\in[0,\infty]$, $r+\phi$ and
$\phi\ominus r$ are the tensor and cotensor of $\phi$ with $r$ in
$(\mathfrak{L}X,\rho_{X})$, respectively; for each $\lambda\in\mathfrak{U}X$
and $r\in[0,\infty]$, $\lambda+r$ and $\lambda\ominus r$ are the tensor and
cotensor of $\lambda$ with $r$ in $(\mathfrak{U}X,\rho_{X})$, respectively. ∎

For each approach space $(X,\delta)$, it is clear that

$d_{\delta}(x,y)\coloneqq\delta(x,\\{y\\})$

is a metric on $X$, called the _specialization metric_ of $(X,\delta)$. We
leave it to the reader to check that every lower regular function of
$(X,\delta)$ is a weight and every upper regular function is a coweight of
$(X,d_{\delta})$.

A function $\lambda\colon X\longrightarrow[0,\infty]$ is said to be
_saturated_ if it is a coweight of the metric space $(X,d_{\delta})$; that is,
$d_{\delta}(x,y)+\lambda(x)\geq\lambda(y)$ for all $x,y\in X$. All upper
regular functions are saturated; we have more.

######  Proposition 3.6.

Let $(X,\delta)$ be an approach space and $\theta\colon
X\longrightarrow[0,\infty]$.

  1. (i)

The function
$\uparrow\\!\theta\coloneqq\sup\\{\lambda\in\mathfrak{U}X\mid\lambda\leq\theta\\}$
is saturated.

  2. (ii)

$\theta$ is saturated if and only if $\theta=\uparrow\\!\theta$.

Thus, $\uparrow\\!\theta$ is called the saturation of $\theta$.

###### Proof.

(i) Trivial, since every upper regular function of $(X,\delta)$ is a coweight
of $(X,d_{\delta})$.

(ii) It suffices to check that if $\theta$ is saturated then
$\theta\leq\uparrow\\!\theta$. Let $x\in X$. If $\theta(x)<\infty$, then
$\theta(x)\ominus\delta(-,\\{x\\})$ is upper regular and
$\theta(x)\ominus\delta(-,\\{x\\})\leq\theta$. So,
$\theta(x)=\theta(x)\ominus\delta(x,\\{x\\})\leq\uparrow\\!\theta(x)$. If
$\theta(x)=\infty$, then for each $r<\infty$, $r\ominus\delta(-,\\{x\\})$ is
upper regular and $r\ominus\delta(-,\\{x\\})\leq\theta$. So
$\uparrow\\!\theta(x)\geq r$, and consequently, $\uparrow\\!\theta(x)=\infty$
by arbitrariness of $r$. ∎

######  Corollary 3.7.

Let $(X,\delta)$ be an approach space and $\theta\in[0,\infty]^{X}$.

  1. (i)

For each upper regular function $\lambda$ of $X$,
$\rho_{X}(\theta,\lambda)=\rho_{X}(\uparrow\\!\theta,\lambda).$

  2. (ii)

$\uparrow\\!\theta=\sup\\{\lambda\ominus\rho_{X}(\theta,\lambda)\mid\lambda\in\mathfrak{U}X\\}.$

######  Definition 3.8.

([1]) A $[0,\infty]$-filter $\lambda$ of a metric space $(X,d)$ is open if
$\lambda\colon(X,d)\longrightarrow([0,\infty],d_{L})$ is $[0,\infty]$-Scott
continuous.

Let $(X,\delta)$ be an approach space and let $\theta\in[0,\infty]^{X}$. By
Proposition 2.11 it is seen readily that $\rho_{X}(\theta,-)$ is a
$[0,\infty]$-filter of the cocomplete metric space $(\mathfrak{U}X,\rho_{X})$.
Now we introduce the central notion of this paper.

######  Definition 3.9.

Let $(X,\delta)$ be an approach space. A function $\theta\colon
X\longrightarrow[0,\infty]$ is compact if $\rho_{X}(\theta,-)$ is an open
$[0,\infty]$-filter of the metric space $(\mathfrak{U}X,\rho_{X})$ of upper
regular functions.

This definition is a direct extension of the characterization of compact
subsets in [15, Corollary 2.14] to the $[0,\infty]$-enriched context.

The following property of compact functions follows immediately from Corollary
3.7.

######  Proposition 3.10.

Let $(X,\delta)$ be an approach space and $\theta\colon
X\longrightarrow[0,\infty]$ be a function. Then $\theta$ is compact if and
only if so is its saturation $\uparrow\\!\theta$.

Since $(\mathfrak{U}X,\rho_{X})$ is a cocomplete metric space and its
underlying order is opposite to the usual pointwise order, from Proposition
2.12 we immediately derive the following characterization of compact
functions.

######  Proposition 3.11.

Let $(X,\delta)$ be a metric space and $\theta\colon
X\longrightarrow[0,\infty]$. Then $\theta$ is a compact function of
$(X,\delta)$ if and only if for each filtered family $\\{\lambda_{i}\\}_{i\in
I}$ of upper regular functions,

$\rho_{X}(\theta,\inf_{i\in I}\lambda_{i})=\inf_{i\in
I}\rho_{X}(\theta,\lambda_{i}).$

The following proposition is an analogy of the fact that continuous images of
compact sets are compact.

######  Proposition 3.12.

Let $f\colon(X,\delta_{X})\longrightarrow(Y,\delta_{Y})$ be a contraction
between approach spaces. Then for each compact function $\theta$ of
$(X,\delta_{X})$, the function

$f(\theta)\colon Y\longrightarrow[0,\infty],\quad
f(\theta)(y)=\inf\\{\theta(x)\mid f(x)=y\\}$

is compact in $(Y,\delta_{Y})$.

###### Proof.

This follows directly from the above proposition and the fact that

$\rho_{Y}(f(\theta),\lambda)=\rho_{X}(\theta,\lambda\circ f)$

for every upper regular function $\lambda$ of $(Y,\delta_{Y})$. ∎

In a topological space $X$, a subset $K$ is compact if and only if for each
filtered family of closed sets $\\{F_{i}\\}_{i\in I}$ such that $K$ meets each
$F_{i}$, $K$ meets the intersection $\bigcap_{i\in I}F_{i}$. The following
proposition is an analogy of this fact in the enriched context.

######  Proposition 3.13.

Let $(X,\delta)$ be an approach space. For each function $\theta\colon
X\longrightarrow[0,\infty]$, the following are equivalent:

  1. (1)

$\theta$ is compact.

  2. (2)

For each directed family $\\{\phi_{j}\\}_{j\in J}$ of lower regular functions,

$(\sup_{j\in J}\phi_{j})\operatorname{\pitchfork}\theta=\sup_{j\in
J}(\phi_{j}\operatorname{\pitchfork}\theta).$

  3. (3)

The map

$-\pitchfork\theta\colon(\mathfrak{L}X,\rho_{X})\longrightarrow([0,\infty],d_{L})$

preserves limits of $[0,\infty]$-filters.

###### Proof.

$(1)\Rightarrow(2)$ It suffices to check that $(\sup_{j\in
J}\phi_{j})\operatorname{\pitchfork}\theta\leq\sup_{j\in
J}(\phi_{j}\operatorname{\pitchfork}\theta)$ for each directed family
$\\{\phi_{j}\\}_{j\in J}$ of lower regular functions. We proceed with two
cases.

If $\sup_{j\in J}\phi_{j}$ is bounded by $b<\infty$, then
$\\{b\ominus\phi_{j}\\}_{j\in J}$ is a filtered family of upper regular
functions of $(X,\delta)$. Appealing to Lemma 2.6 we have

$\displaystyle b\ominus((\sup_{j\in
J}\phi_{j})\operatorname{\pitchfork}\theta)$
$\displaystyle=\rho_{X}(\theta,b\ominus\sup_{j\in J}\phi_{j})$
$\displaystyle=\rho_{X}(\theta,\inf_{j\in J}(b\ominus\phi_{j}))$
$\displaystyle=\inf_{j\in J}\rho_{X}(\theta,b\ominus\phi_{j})$
$\displaystyle=\inf_{j\in J}b\ominus(\phi_{j}\operatorname{\pitchfork}\theta)$
$\displaystyle=b\ominus\sup_{j\in
J}(\phi_{j}\operatorname{\pitchfork}\theta).$

From the arbitrariness of $b$ it follows that $(\sup_{j\in
J}\phi_{j})\operatorname{\pitchfork}\theta=\sup_{j\in
J}(\phi_{j}\operatorname{\pitchfork}\theta)$.

If $\sup_{j\in I}\phi_{j}$ is not bounded, then

$\displaystyle(\sup_{j\in J}\phi_{j})\operatorname{\pitchfork}\theta$
$\displaystyle=(\sup_{r\in[0,\infty)}\inf\\{r_{X},\sup_{j\in
J}\phi_{j}\\})\operatorname{\pitchfork}\theta$
$\displaystyle=\sup_{r\in[0,\infty)}(\inf\\{r_{X},\sup_{j\in
J}\phi_{j}\\}\operatorname{\pitchfork}\theta)$
$\displaystyle=\sup_{r\in[0,\infty)}(\sup_{j\in
J}\inf\\{r_{X},\phi_{j}\\})\operatorname{\pitchfork}\theta$
$\displaystyle\leq\sup_{r\in[0,\infty)}\sup_{j\in
J}(\inf\\{r_{X},\phi_{j}\\}\operatorname{\pitchfork}\theta)$
$\displaystyle=\sup_{j\in
J}\sup_{r\in[0,\infty)}(\inf\\{r_{X},\phi_{j}\\}\operatorname{\pitchfork}\theta)$
$\displaystyle=\sup_{j\in J}(\phi_{j}\operatorname{\pitchfork}\theta).$

$(2)\Rightarrow(1)$ Let $\\{\lambda_{i}\\}_{i\in I}$ be a filtered family of
upper regular functions. Without loss of generality, we may suppose that all
$\lambda_{i}$ are bounded by $b<\infty$. Then $\\{b\ominus\lambda_{i}\\}_{i\in
I}$ is a directed family of lower regular functions. Hence by Lemma 2.6 we
have

$\displaystyle\rho_{X}(\theta,\inf_{i\in I}\lambda_{i})$
$\displaystyle=b\ominus((b\ominus\inf_{i\in
I}\lambda_{i})\operatorname{\pitchfork}\theta)$
$\displaystyle=b\ominus((\sup_{i\in
I}(b\ominus\lambda_{i}))\operatorname{\pitchfork}\theta)$
$\displaystyle=b\ominus\sup_{i\in
I}((b\ominus\lambda_{i})\operatorname{\pitchfork}\theta)$
$\displaystyle=\inf_{i\in
I}b\ominus((b\ominus\lambda_{i})\operatorname{\pitchfork}\theta)$
$\displaystyle=\inf_{i\in I}\rho_{X}(\theta,\lambda_{i}),$

which shows that $\theta$ is compact.

$(2)\Leftrightarrow(3)$ This follows from the dual of Proposition 2.9. ∎

######  Proposition 3.14.

Let $(X,\delta)$ be an approach space and $A\subseteq X$. Then a function
$\theta\colon A\longrightarrow[0,\infty]$ is compact in the subspace
$(A,\delta|A)$ if and only if

$\theta^{*}\colon
X\longrightarrow[0,\infty],\quad\theta^{*}(x)=\begin{cases}\theta(x),&x\in
A\\\ \infty,&x\notin A\end{cases}$

is compact in $(X,\delta)$.

###### Proof.

This follows from Proposition 3.13 and the fact that $\lambda\colon
A\longrightarrow[0,\infty]$ is a lower regular function of $(A,\delta|A)$ if
and only if there is some lower regular function $\widetilde{\lambda}$ of
$(X,\delta)$ such that $\lambda$ is the restriction of $\widetilde{\lambda}$
on $A$. ∎

######  Proposition 3.15.

Let $(X,\delta)$ be an approach space. If both $\theta_{1}$ and $\theta_{2}$
are compact functions of $(X,\delta)$, then so is
$\inf\\{\theta_{1},\theta_{2}\\}$.

###### Proof.

Use the fact that for any upper regular function $\lambda$ of $(X,\delta)$,
$\rho_{X}(\inf\\{\theta_{1},\theta_{2}\\},\lambda)=\max\\{\rho_{X}(\theta_{1},\lambda),\rho_{X}(\theta_{2},\lambda)\\}$.
∎

The following proposition implies that the family of compact functions is
determined by the inhabited ones. We note that for each approach space
$(X,\delta)$, the function $\infty_{X}$ is compact.

######  Proposition 3.16.

Let $(X,\delta)$ be an approach space and $\theta\colon
X\longrightarrow[0,\infty]$ be a function.

  1. (i)

If $\theta$ is compact then so is $\theta+r$ for each $r\in[0,\infty]$.

  2. (ii)

If $\theta$ is compact and $\theta\neq\infty_{X}$, then for each
$r\leq\inf\limits_{x\in X}\theta(x)$, $\theta\ominus r$ is compact.

Hence for each $\theta\neq\infty_{X}$, $\theta$ is compact if and only if
$\theta\ominus\inf_{x\in X}\theta(x)$ is compact.

###### Proof.

(i) Since for each directed family $\\{\phi_{i}\\}_{i\in I}$ of lower regular
functions,

$\displaystyle(\sup_{i\in I}\phi_{i})\operatorname{\pitchfork}(\theta+r)$
$\displaystyle=\inf_{x\in X}(\sup_{i\in I}\phi_{i}(x)+\theta(x)+r)$
$\displaystyle=(\sup_{i\in I}\phi_{i})\operatorname{\pitchfork}\theta+r$
$\displaystyle=\sup_{i\in I}(\phi_{i}\operatorname{\pitchfork}\theta)+r$
$\displaystyle=\sup_{i\in I}(\phi_{i}\operatorname{\pitchfork}(\theta+r)),$

hence $\theta+r$ is compact.

(ii) Similar.∎

In [21], Lowen introduced the notion of _index of compactness_ for approach
spaces. In order to discuss the relationship between compact functions and the
compactness-indices of approach spaces, we introduce the following:

######  Definition 3.17.

Let $(X,\delta)$ be an approach space and $\theta\in[0,\infty]^{X}$. The value

$\chi(\theta)\coloneqq\sup_{I}\Big{(}(\sup_{\phi\in
I}\phi)\pitchfork\theta\ominus\sup_{\phi\in I}(\phi\pitchfork\theta)\Big{)}$

is called the index of compactness of $\theta$, where $I$ ranges over all
ideals of the complete lattice $\mathfrak{L}X$ of lower regular functions of
$(X,\delta)$.

We leave it to the reader to check that

$\chi(\theta)=\sup_{F}\Big{(}\Big{(}\inf_{\lambda\in
F}\rho_{X}(\theta,\lambda)\Big{)}\ominus\rho_{X}(\theta,\inf_{\lambda\in
F}\lambda)\Big{)},$

where $F$ ranges over all filters of the complete lattice $\mathfrak{U}X$ of
upper regular functions of $(X,\delta)$.

It is clear that $\theta$ is a compact function if and only if
$\chi(\theta)=0$; that is, $\theta$ has $0$ as index of compactness.

######  Proposition 3.18.

Let $(X,\delta)$ be an approach space.

  1. (i)

The index of compactness of the constant function $0_{X}$ is given by

$\chi(0_{X})=\sup_{J}\inf_{x\in X}\sup_{\phi\in J}\phi(x),$

where $J$ ranges over all ideals composed of inhabited lower regular
functions.

  2. (ii)

$\chi(0_{X})$ is equal to the index of compactness of $(X,\delta)$ in the
sense of Lowen [21, 24].

  3. (iii)

$\chi(0_{X})=0$ if and only if $(X,\delta)$ is $0$-compact in the sense of
Lowen [24].

###### Proof.

(i) By definition,

$\chi(0_{X})=\sup_{I}\Big{(}\inf_{x\in X}\sup_{\phi\in
I}\phi(x)\ominus\sup_{\phi\in I}\inf_{x\in X}\phi(x)\Big{)},$

where $I$ ranges over all ideals of the complete lattice $\mathfrak{L}X$.
Since $\sup_{\phi\in J}\inf_{x\in X}\phi(x)=0$ for each ideal $J$ composed of
inhabited lower regular functions, it suffices to check that for each ideal
$I$ of $\mathfrak{L}X$, there is an ideal $J$ composed of inhabited lower
regular functions such that

$\inf_{x\in X}\sup_{\phi\in I}\phi(x)\ominus\sup_{\phi\in I}\inf_{x\in
X}\phi(x)\leq\inf_{x\in X}\sup_{\psi\in J}\psi(x).$

If $\sup_{\phi\in I}\inf_{x\in X}\phi(x)=\infty$, any ideal composed of
inhabited lower regular functions meets the requirement. If $\sup_{\phi\in
I}\inf_{x\in X}\phi(x)=b<\infty$, then $\\{\phi\ominus b\\}_{\phi\in I}$ is a
directed family consisting of inhabited lower regular functions. Let $J$ be
the ideal generated by this family. Then $J$ is composed of inhabited lower
regular functions and

$\inf_{x\in X}\sup_{\psi\in J}\psi(x)=\inf_{x\in X}\sup_{\phi\in
I}(\phi(x)\ominus b)=\inf_{x\in X}\sup_{\phi\in I}\phi(x)\ominus b,$

which shows that $J$ satisfies the requirement.

(ii) Follows from (i) and [24, Theorem 4.3.9].

(iii) A special case of (ii). ∎

The above proposition shows that Definition 3.17 provides an extension of the
notion of index of compactness for approach spaces in the sense of Lowen; and
it also leads to the following:

######  Definition 3.19.

We say that an approach space $(X,\delta)$ is compact if its index of
compactness is $0$; or equivalently, $0_{X}$ is a compact function.

Compact approach spaces are just the $0$-compact ones in the sense of Lowen
[24]. It follows from Lemma 3.16 that an approach space $(X,\delta)$ is
compact if and only if for all $r\in[0,\infty]$, the constant function $r_{X}$
is compact.

In the following we show that compact functions in approach spaces are an
_extension_ of compact subsets in topological spaces and precompact subsets in
metric spaces.

For each topological space $X$, the map

$\delta_{X}\colon X\times
2^{X}\longrightarrow[0,\infty],\quad\delta_{X}(x,A)=\begin{cases}0,&\text{if
$x$ is in the closure of $A$},\\\ \infty,&\text{otherwise}\end{cases}$

is an approach structure on the set $X$. The correspondence
$X\mapsto\omega(X)\coloneqq(X,\delta_{X})$ defines a full and faithful functor

$\omega\colon\sf Top\longrightarrow\sf App.$

Approach spaces of the form $\omega(X)$ are said to be _topologically
generated_ [23, 24]. It is clear that a function $\phi\colon
X\longrightarrow[0,\infty]$ is a lower regular function of $\omega(X)$ if and
only if $\phi$ is lower semicontinuous; and that $\lambda\colon
X\longrightarrow[0,\infty]$ is an upper regular function of $\omega(X)$ if and
only if either $\lambda=\infty_{X}$ or $\lambda$ is bounded and upper
semicontinuous.

For a topological space $X$, it is shown in [24, Proposition 4.3.11] that the
approach space $\omega(X)$ is compact if and only if the topological space $X$
is compact. This fact together with Proposition 3.14 yields the following
proposition, which shows that the notion of compact functions of approach
spaces extends that of compact subsets of topological spaces.

######  Proposition 3.20.

Let $X$ be a topological space and $A$ a subset of $X$. Then $0_{A}$ is a
compact function of $\omega(X)$ if and only if $A$ is a compact subset of $X$.

######  Proposition 3.21.

Let $X$ be a topological space and let $\theta\colon
X\longrightarrow[0,\infty]$ be a lower semicontinuous function. Then $\theta$
is a compact function of $\omega(X)$ if and only if $\theta^{-1}([0,r])$ is a
compact subset of $X$ for each $r\in[0,\infty)$.

###### Proof.

We check the necessity first. Let $A=\theta^{-1}[0,r]$. Since $\theta$ is
lower semicontinuous, $A$ is a closed set in X. Let $\\{F_{i}\\}_{i\in I}$ be
a filtered family of non-empty closed sets of $A$. Since $A$ is closed, it
follows that $\\{0_{F_{i}}\\}_{i\in I}$ is a directed family of lower regular
functions of $\omega(X)$, hence

$(\sup_{i\in I}0_{F_{i}})\operatorname{\pitchfork}\theta=\sup_{i\in
I}(0_{F_{i}}\operatorname{\pitchfork}\theta)\leq r,$

which shows that $\\{F_{i}\\}_{i\in I}$ has a nonempty intersection and
consequently, $A$ is compact.

As for sufficiency, it suffices to prove that, for each directed family of
lower semicontinuous functions $\\{\phi_{i}\colon
X\longrightarrow[0,\infty]\\}_{i\in I}$,

$(\sup_{i\in I}\phi_{i})\operatorname{\pitchfork}\theta\leq\sup_{i\in
I}(\phi_{i}\operatorname{\pitchfork}\theta).$

If $\sup_{i\in I}(\phi_{i}\operatorname{\pitchfork}\theta)<r<\infty$, then for
each $i\in I$,

$\inf_{x\in X}(\phi_{i}(x)+\theta(x))<r.$

Let $F_{i}=(\phi_{i}+\theta)^{-1}[0,r]$ for each $i\in I$. By lower
semicontinuity of $\theta$ and $\phi_{i}$, one sees that $\\{F_{i}\\}_{i\in
I}$ is a filtered family of non-empty closed sets contained in the closed set
$A\coloneqq\theta^{-1}[0,r]$. Since $A$ is compact, then

$(\sup_{i\in I}\phi_{i})\operatorname{\pitchfork}\theta=\inf_{x\in
X}(\sup_{i\in I}\phi_{i}(x)+\theta(x))\leq r$

and consequently,

$(\sup_{i\in I}\phi_{i})\operatorname{\pitchfork}\theta\leq\sup_{i\in
I}(\phi_{i}\operatorname{\pitchfork}\theta)$

as desired. ∎

######  Corollary 3.22.

If $X$ is a compact topological space, then every lower semicontinuous
function from $X$ to $[0,\infty]$ is compact in $\omega(X)$.

Given a metric space $(X,d)$, the map

$\Gamma(d)\colon X\times
2^{X}\longrightarrow[0,\infty],\quad\Gamma(d)(x,A)=\begin{cases}\infty,&A=\varnothing,\\\
\inf\limits_{y\in A}d(x,y),&A\neq\varnothing\end{cases}$

is an approach structure on the set $X$. Spaces of the form $(X,\Gamma(d))$
are called _metric approach spaces_ [23, 24].

It is not hard to check that $\phi\colon X\longrightarrow[0,\infty]$ is a
lower regular function of the metric approach space $(X,\Gamma(d))$ if and
only if $\phi$ is a weight of $(X,d)$; and $\lambda\colon
X\longrightarrow[0,\infty]$ is an upper regular function of $(X,\Gamma(d))$ if
and only if either $\lambda=\infty_{X}$ or $\lambda$ is a bounded coweight of
$(X,d)$. Then, by the dual statement of the Yoneda lemma (Lemma 2.1) one sees
that for each $x\in X$, $d(x,-)\colon X\longrightarrow[0,\infty]$ is a compact
function of $(X,\Gamma(d))$. So, by Proposition 3.15, for each finite subset
$F$ of $X$,

$d(F,-)\coloneqq\inf_{a\in F}d(a,-)$

is a compact function of $(X,\Gamma(d))$.

A metric space $(X,d)$ is _precompact_ if for each $r>0$, there exists a
finite subset $A$ of $X$, such that $X=\bigcup_{x\in A}B(x,r),$ where
$B(x,r)=\\{y\in X\mid d(x,y)<r\\}$. A subset $K$ is called a precompact subset
if $(K,d)$ is a precompact metric space.

######  Proposition 3.23.

Let $(X,d)$ be a metric space and $K$ a subset of $X$. The following are
equivalent:

  1. (1)

$K$ is a precompact subset of $(X,d)$.

  2. (2)

$d(K,-)$ is a compact function of $(X,\Gamma(d))$.

  3. (3)

$0_{K}$ is a compact function of $(X,\Gamma(d))$.

######  Lemma 3.24.

Let $(X,\delta)$ be an approach space and $\theta\colon
X\longrightarrow[0,\infty]$ be a function. If for each $r>0$, there exists a
compact function $\xi$ such that $\theta\leq\xi\leq\theta+r$, then $\theta$ is
compact.

###### Proof.

Let $\\{\phi_{i}\\}_{i\in I}$ be a directed family of lower regular functions
and let $r>0$. Pick a compact function $\xi$ such that
$\theta\leq\xi\leq\theta+r$. Then

$\displaystyle(\sup_{i\in I}\phi_{i})\operatorname{\pitchfork}\theta$
$\displaystyle\leq(\sup_{i\in I}\phi_{i})\operatorname{\pitchfork}\xi$
$\displaystyle=\sup_{i\in I}(\phi_{i}\operatorname{\pitchfork}\xi)$
$\displaystyle\leq\sup_{i\in I}(\phi_{i}\operatorname{\pitchfork}(\theta+r))$
$\displaystyle=\sup_{i\in I}(\phi_{i}\operatorname{\pitchfork}\theta)+r,$

hence $\theta$ is compact by arbitrariness of $r$. ∎

###### Proof of Proposition 3.23.

First, it is shown in [21, Theorem 4.4] that for a metric space $(X,d)$, the
approach space $(X,\Gamma(d))$ is compact if and only if $(X,d)$ is
precompact. The equivalence $(1)\Leftrightarrow(3)$ follows immediately from
this fact and Proposition 3.14. Here we present a proof of the proposition by
help of Lemma 3.24.

$(1)\Rightarrow(2)$ Since $K$ is a precompact subset of $X$, for each $r>0$,
there exists a finite subset $F$ of $K$ such that,

$K\subseteq\bigcup_{x\in F}B(x,r).$

It follows that

$d(K,y)\leq d(F,y)<d(K,y)+r$

for each $y\in X$. Since $F$ is finite, $d(F,-)$ is a compact function, so,
$d(K,-)$ is compact by Lemma 3.24.

$(2)\Rightarrow(3)$ This follows from that for each weight $\phi$ of $(X,d)$,

$\displaystyle\phi\operatorname{\pitchfork}d(K,-)$ $\displaystyle=\inf_{x\in
X}(\phi(x)+d(K,x))$ $\displaystyle=\inf_{x\in K}\phi(x)$
$\displaystyle=\phi\operatorname{\pitchfork}0_{K}.$

$(3)\Rightarrow(1)$ Suppose on the contrary that $K$ is not precompact. Then
there exists $r>0$ such that $K\nsubseteq\bigcup_{x\in F}B(x,r)$ for any
finite subset $F$ of $K$. For each $x\in K$, let

$\phi_{x}=d(-,X\backslash B(x,r)).$

Since $\phi_{x}(x)\geq r$ for all $x\in K$, it follows that

$(\sup_{x\in K}\phi_{x})\operatorname{\pitchfork}0_{K}=\inf_{y\in K}\sup_{x\in
K}\phi_{x}(y)\geq r.$

For each finite subset $F$ of $K$, there is some $z\in K$ such that
$d(x,z)\geq r$ for all $x\in F$. Thus,

$(\sup_{x\in F}\phi_{x})\operatorname{\pitchfork}0_{K}=\inf_{y\in K}\sup_{x\in
F}\phi_{x}(y)\leq\sup_{x\in F}\phi_{x}(z)=0,$

which contradicts the compactness of $0_{K}$. ∎

##  4 Hofmann-Mislove theorem

######  Definition 4.1.

Let $(X,\delta)$ be an approach space and let $\Lambda$ be a
$[0,\infty]$-filter of the metric space $(\mathfrak{U}X,\rho_{X})$ of upper
regular functions of $(X,\delta)$. We say that $\Lambda$ is

  1. (i)

decent if $\Lambda(\lambda)\neq 0$ for some $\lambda\neq\infty_{X}$;

  2. (ii)

proper if $\Lambda(r_{X})=r$ for each $r\in[0,\infty]$.

For each approach space $(X,\delta)$ with $X$ being non-empty, the metric
space $(\mathfrak{U}X,\rho_{X})$ has exactly two $[0,\infty]$-filters that are
not decent, both of them are open. The first is $\rho_{X}(\infty_{X},-)$ and
the second is given by

$\Lambda(\lambda)=\begin{cases}0,&\lambda\neq\infty_{X},\\\
\infty,&\lambda=\infty_{X}.\end{cases}$

The following proposition is obvious.

######  Proposition 4.2.

Let $\theta$ be a compact saturated function of an approach space
$(X,\delta)$.

  1. (i)

If $\theta$ is inhabited then $\rho_{X}(\theta,-)$ is a proper open
$[0,\infty]$-filter of $(\mathfrak{U}X,\rho_{X})$.

  2. (ii)

If $\theta$ is non-vacuous then $\rho_{X}(\theta,-)$ is a decent open
$[0,\infty]$-filter of $(\mathfrak{U}X,\rho_{X})$.

Our aim is to show that for a sober approach space $(X,\delta)$, every proper
open $[0,\infty]$-filter of $(\mathfrak{U}X,\rho_{X})$ is of the form
$\rho_{X}(\theta,-)$ for a unique inhabited and saturated compact function.

A lower regular function $\phi$ of an approach space $(X,\delta)$ is said to
be _irreducible_ if for all lower regular functions $\phi_{1}$ and $\phi_{2}$,

$\inf\\{\phi_{1},\phi_{2}\\}\leq\phi\implies\text{ either $\phi_{1}\leq\phi$
or $\phi_{2}\leq\phi$}.$

######  Definition 4.3.

[2] An approach space $(X,\delta)$ is sober if for each inhabited and
irreducible lower regular function $\phi$ of $(X,\delta)$, there is a unique
$x\in X$ such that $\phi=\delta(-,\\{x\\})$.

######  Example 4.4.

  1. (a)

([2]) For each topological space $X$, the approach space $\omega(X)$ is sober
if and only if $X$, as a topological space, is sober.

  2. (b)

([19]) For each metric space $(X,d)$, the metric approach space
$(X,\Gamma(d))$ is sober if and only if $(X,d)$ is Smyth complete; that is,
$(X,d)$ is separated and all of its forward Cauchy nets converge in its
symmetrization (defined in an obvious way). In the case that the metric $d$ is
symmetric, this conclusion was first proved in [2].

######  Proposition 4.5.

An approach space $(X,\delta)$ is sober if and only if for each non-vacuous
irreducible lower regular function $\phi$ of $(X,\delta)$, there exist a
unique $x\in X$ and a unique $s\in[0,\infty)$ such that
$\phi=s+\delta(-,\\{x\\})$.

###### Proof.

Consider the lower regular function $\phi\ominus\inf_{x\in X}\phi(x)$.∎

Now we state the main result of this paper.

######  Theorem 4.6.

Let $(X,\delta)$ be a sober approach space.

  1. (i)

For each proper open $[0,\infty]$-filter $\Lambda$ of
$(\mathfrak{U}X,\rho_{X})$, there is a unique inhabited compact saturated
function $\nu$ of $(X,\delta)$ such that $\Lambda=\rho_{X}(\nu,-)$.

  2. (ii)

For each decent open $[0,\infty]$-filter $\Lambda$ of
$(\mathfrak{U}X,\rho_{X})$, there is a unique non-vacuous compact saturated
function $\mu$ of $(X,\delta)$ such that $\Lambda=\rho_{X}(\mu,-)$.

Therefore, for a sober approach space $(X,\delta)$, the proper (decent, resp.)
open $[0,\infty]$-filters of the metric space $(\mathfrak{U}X,\rho_{X})$ of
upper regular functions correspond bijectively to the inhabited (non-vacuous,
resp.) compact saturated functions of $(X,\delta)$.

A lemma first.

######  Lemma 4.7.

Let $(X,\delta)$ be an approach space. If $\lambda$ is a compact saturated
function of $(X,\delta)$, then the map

$\Phi\colon\mathfrak{L}X\longrightarrow[0,\infty],\quad\Phi(\phi)=\phi\operatorname{\pitchfork}\lambda$

satisfies the following conditions:

  1. (C1)

For each directed subset $\\{\phi_{i}\\}_{i}$ of $\mathfrak{L}X$,
$\Phi(\sup\limits_{i\in I}\phi_{i})=\sup\limits_{i\in I}\Phi(\phi_{i})$.

  2. (C2)

For all $\phi_{1},\phi_{2}\in\mathfrak{L}X$,
$\Phi(\inf\\{\phi_{1},\phi_{2}\\})=\min\\{\Phi(\phi_{1}),\Phi(\phi_{2})\\}$.

  3. (C3)

For all $\phi\in\mathfrak{L}X$ and $r\in[0,\infty]$,
$\Phi(\phi+r)=\Phi(\phi)+r$.

Conversely, if $(X,\delta)$ is a sober approach space, then for each map
$\Phi\colon\mathfrak{L}X\longrightarrow[0,\infty]$ that satisfies (C1)-(C3),
there is a compact saturated function $\lambda$ of $(X,\delta)$ such that
$\Phi(\phi)=\phi\operatorname{\pitchfork}\lambda$.

###### Proof.

We leave it to the reader to check that for each compact saturated function
$\lambda$ of $(X,\delta)$, the map
$\Phi(\phi)=\phi\operatorname{\pitchfork}\lambda$ satisfies (C1)-(C3).

For the converse part, suppose that
$\Phi\colon\mathfrak{L}X\longrightarrow[0,\infty]$ satisfies (C1)-(C3).

First of all, by (C2), (C3) and Proposition 2.4 one sees that
$\Phi\colon(\mathfrak{L}X,\rho_{X})\longrightarrow([0,\infty],d_{L})$ is a
non-expansive map. We claim that the function

$\lambda\colon
X\longrightarrow[0,\infty],\quad\lambda(x)=\Phi(\delta(-,\\{x\\}))$

satisfies the requirement; that is, $\lambda$ is a compact saturated function
of $(X,\delta)$ and $\Phi(\phi)=\phi\operatorname{\pitchfork}\lambda$.

Since

$\lambda(x)+\delta(x,\\{y\\})=\Phi(\delta(-,\\{x\\}))+\rho_{X}(\delta(-,\\{x\\}),\delta(-,\\{y\\}))\geq\Phi(\delta(-,\\{y\\}))=\lambda(y),$

it follows that $\lambda$ is saturated.

Next we show that for each $\phi\in\mathfrak{L}X$,
$\Phi(\phi)=\phi\operatorname{\pitchfork}\lambda$. Note that this equality
together with (C1) imply that $\lambda$ is compact. On one hand,

$\displaystyle\phi\operatorname{\pitchfork}\lambda$ $\displaystyle=\inf_{x\in
X}(\phi(x)+\Phi(\delta(-,\\{x\\})))$ $\displaystyle=\inf_{x\in
X}(\rho_{X}(\delta(-,\\{x\\}),\phi)+\Phi(\delta(-,\\{x\\})))$ (Equation (3.i))
$\displaystyle\geq\Phi(\phi).$

On the other hand, for each $r\in[0,\infty)$ with $\Phi(\phi)\leq r$, let

$\mathcal{A}_{r}=\\{\xi\in\mathfrak{L}X\mid\phi\leq\xi,\Phi(\xi)\leq r\\}.$

By (C1) and Zorn’s lemma, $\mathcal{A}_{r}$ has a maximal element, say $\psi$.
By (C2) one sees that $\psi$ is an irreducible lower regular function; by (C3)
one sees that $\psi$ is non-vacuous. Since $(X,\delta)$ is sober, there exist
a unique $x$ and a unique $s\in[0,\infty)$ such that
$\psi=s+\delta(-,\\{x\\})$. Since $\psi\in\mathcal{A}_{r}$, it follows that

$\displaystyle\phi\operatorname{\pitchfork}\lambda$
$\displaystyle\leq\psi\operatorname{\pitchfork}\lambda$
$\displaystyle=\inf_{z\in X}(s+\delta(z,\\{x\\})+\Phi(\delta(-,\\{z\\})))$
$\displaystyle=s+\inf_{z\in
X}\Big{(}\rho_{X}(\delta(-,\\{z\\}),\delta(-,\\{x\\}))+\Phi(\delta(-,\\{z\\}))\Big{)}$
$\displaystyle=s+\Phi(\delta(-,\\{x\\}))$ ($\Phi$ is non-expansive)
$\displaystyle=\Phi(\psi)$ (C3) $\displaystyle\leq r.$

Therefore, $\phi\operatorname{\pitchfork}\lambda\leq\Phi(\phi)$ by
arbitrariness of $r$. ∎

###### Proof of Theorem 4.6.

(i) Suppose that $\Lambda\colon\mathfrak{U}X\longrightarrow[0,\infty]$ is a
proper open $[0,\infty]$-filter of $(\mathfrak{U}X,\rho_{X})$. First of all,
we assert that $\Lambda$ satisfies that

$\Lambda(\lambda+r)=\Lambda(\lambda)+r$

for all $\lambda\in\mathfrak{U}X$ and $r\in[0,\infty]$. If
$\lambda=\infty_{X}$, the equality is obvious; if $\lambda$ is bounded, then
by Proposition 2.11 we have that $\Lambda(\lambda+r)\geq r$ and

$\Lambda(\lambda)=\Lambda((\lambda+r)\ominus r)=\Lambda(\lambda+r)-r,$

hence $\Lambda(\lambda+r)=\Lambda(\lambda)+r$.

We turn to the proof of the conclusion. Uniqueness of $\nu$ follows from
Corollary 3.7; as for existence, define a map
$\Phi\colon\mathfrak{L}X\longrightarrow[0,\infty]$ as follows: for each lower
regular function $\phi$ of $(X,\delta)$, let

$\Phi(\phi)=\sup_{x\in X}\phi(x)\ominus\Lambda\Big{(}\sup_{x\in
X}\phi(x)\ominus\phi\Big{)}$

if $\phi$ is bounded; and let

$\Phi(\phi)=\sup_{r\in[0,\infty)}\Phi(\inf\\{\phi,r_{X}\\})$

if $\phi$ is not bounded.

Before proceeding, we notice that, as a consequence of the equality
$\Lambda(\lambda+r)=\Lambda(\lambda)+r$, for each bounded $\phi$ and each
bound $b$ of $\phi$, it holds that

$\Phi(\phi)=b-\Lambda(b-\phi).$

We claim that $\Phi\colon\mathfrak{L}X\longrightarrow[0,\infty]$ satisfies the
conditions (C1)-(C3) in Lemma 4.7. Here we check (C1) for example. Let
$\\{\phi_{i}\\}_{i\in I}$ be a directed family of lower regular functions. If
$\sup_{i\in I}\phi_{i}$ is bounded, pick an upper bound $b$ of all $\phi_{i}$,
then

$\displaystyle\Phi(\sup_{i\in I}\phi_{i})$
$\displaystyle=b\ominus\Lambda(b\ominus\sup_{i\in I}\phi_{i})$
$\displaystyle=b\ominus\Lambda(\inf_{i\in I}b\ominus\phi_{i})$
$\displaystyle=b\ominus\inf_{i\in I}\Lambda(b\ominus\phi_{i})$
$\displaystyle=\sup_{i\in I}b\ominus\Lambda(b\ominus\phi_{i})$
$\displaystyle=\sup_{i\in I}\Phi(\phi_{i}).$

If $\sup_{i\in I}\phi_{i}$ is not bounded, we have

$\displaystyle\Phi(\sup_{i\in I}\phi_{i})$
$\displaystyle=\sup_{r\in[0,\infty)}\Phi(\inf\\{\sup_{i\in
I}\phi_{i},r_{X}\\})$ $\displaystyle=\sup_{r\in[0,\infty)}\Phi(\sup_{i\in
I}\inf\\{\phi_{i},r_{X}\\})$ $\displaystyle=\sup_{r\in[0,\infty)}\sup_{i\in
I}\Phi(\inf\\{\phi_{i},r_{X}\\})$ $\displaystyle=\sup_{i\in
I}\sup_{r\in[0,\infty)}\Phi(\inf\\{\phi_{i},r_{X}\\})$
$\displaystyle=\sup_{i\in I}\Phi(\phi_{i}).$

Since $(X,\delta)$ is sober, by Lemma 4.7, there is a compact saturated
function $\nu$ of $(X,\delta)$ such that
$\Phi(\phi)=\phi\operatorname{\pitchfork}\nu$. We show that $\nu$ satisfies
the requirement. First, it follows from
$0_{X}\operatorname{\pitchfork}\nu=\Phi(0_{X})=0$ that $\nu$ is inhabited.
Second, we show that $\rho_{X}(\nu,\lambda)=\Lambda(\lambda)$ for all
$\lambda\in\mathfrak{U}X$. The equality is trivial if $\lambda=\infty_{X}$; if
$\lambda$ is bounded, pick a bound $b$ of $\lambda$, then

$\displaystyle\rho_{X}(\nu,\lambda)$
$\displaystyle=\rho_{X}(\nu,b\ominus(b\ominus\lambda))$
$\displaystyle=b\ominus((b\ominus\lambda)\operatorname{\pitchfork}\nu)$ (Lemma
2.6) $\displaystyle=b\ominus\Phi(b\ominus\lambda)$
$\displaystyle=b\ominus(b-\Lambda(b-(b-\lambda)))$ ($b$ is a bound of
$b-\lambda$) $\displaystyle=b-(b-\Lambda(\lambda))$
$\displaystyle(\Lambda(\lambda)\leq b)$ $\displaystyle=\Lambda(\lambda).$

(ii) Suppose $\Lambda\colon\mathfrak{U}X\longrightarrow[0,\infty]$ is a decent
open $[0,\infty]$-filter of $(\mathfrak{U}X,\rho_{X})$. Let

$s=\sup\\{r\in[0,\infty)\mid\Lambda(r_{X})=0\\}.$

Since $\Lambda$ is decent, $s<\infty$. Then for each $r\in[0,\infty)$,
$\Lambda(r_{X}+s)=r$. Hence the map

$\Lambda^{\prime}\colon\mathfrak{U}X\longrightarrow[0,\infty],\quad\Lambda^{\prime}(\lambda)=\Lambda(\lambda+s)$

is a proper open $[0,\infty]$-filter of $(\mathfrak{U}X,\rho_{X})$. By (i),
there exists a unique inhabited compact saturated function $\nu$ of
$(X,\delta)$ such that $\Lambda^{\prime}=\rho_{X}(\nu,-)$. Since for each
$\lambda\in\mathfrak{U}X$ we have

$\displaystyle\Lambda(\lambda)$ $\displaystyle=\Lambda((\lambda+s)\ominus s)$
$\displaystyle=\Lambda(\lambda+s)\ominus s$
$\displaystyle=\Lambda^{\prime}(\lambda)\ominus s$
$\displaystyle=\rho_{X}(\nu,\lambda)\ominus s$
$\displaystyle=\rho_{X}(\nu+s,\lambda),$

then $\mu\coloneqq\nu+s$ is a compact saturated function of $(X,\delta)$ such
that $\Lambda=\rho_{X}(\mu,-)$. ∎

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