# Strongly Continuous Domains

Yinglong Song  Jinbo Yang  School of Mathematics and Statistics, Jiangxi
Normal University, Nanchang, Jiangxi 330022, People’s Republic of China
Jiangxi Provincial Center for Applied Mathematics, Jiangxi Normal University,
Nanchang, Jiangxi 330022, People’s Republic of China

###### Abstract

Strong Scott topology introduced by X. Xu and D. Zhao is a kind of new
topology which is finer than upper topology and coarser than Scott topology.
Inspired by the topological characterizations of continuous domains and
hypercontinuous domains, we introduce the concept of strongly continuous
domains and investigate some properties of strongly continuous domains. In
particular, we give the definition of strong way-below relation and obtain a
characterization of strongly continuous domains via the strong way-below
relation. We prove that the strong way-below relation on a strongly continuous
domain satisfies the interpolation property, and clarify the relationship
between strongly continuous domains and continuous domains, and the
relationship between strongly continuous domains and hypercontinuous domains.
We discuss the properties of strong Scott topology and strong Lawson topology,
which is the common refinement of the strong Scott topology and the lower
topology, on a strongly continuous domain.

###### keywords:

strongly continuous domains, strong way-below relation, strong Scott topology,
strong Lawson topology

††journal: Electronic Notes in Theoretical Informatics and Computer
Science††volume: 2††thanks: Supported by the NSF of China (12071188, 11361028,
11671008) and Science and Technology Project from Jiangxi Education Department
(GJJ150344)††thanks: Email<EMAIL_ADDRESS>Email:
<EMAIL_ADDRESS>
##  1 Introduction

In non-Hausdorff topology and domain theory, $d$-spaces, well-filtered spaces
and sober spaces form three of the most important classes of $T_{0}$ spaces.
In the past few years, some remarkable progresses have been achieved in the
research of these three kinds of spaces(see, e.g., [6, 7, 8, 9, 11]). In order
to reveal finer links between well-filtered spaces and $d$-spaces, X. Xu and
D. Zhao introduced another class of $T_{0}$ spaces– _strong $d$-spaces_ which
lie between $d$-spaces and $T_{1}$ spaces in [8]. It is well-known that a
$T_{0}$ space $X$ is a $d$-space iff $X$ is a _directed complete poset_
(_dcpo_ , for short) and the topology on $X$ is coarser than the Scott
topology on $X$ with respect to its _specialization order_. In order to obtain
some characterizations of strong $d$-spaces ([8, Proposition 3.22]), the
authors [8] introduced the concept of _strong Scott topology_ and showed that
for a $T_{0}$ space $X$, if $X$ is a strong $d$-space, then $X$ is a $d$-space
and the topology on $X$ is coarser than the strong Scott topology on $X$, and
the converse is true if $X$ is a sup semilattice with the specialization
order.

Strong Scott topology is a kind of new topology which is finer than the upper
topology and coarser than the Scott topology. Inspired by the topological
characterizations of continuous domains (resp., hypercontinuous domains) that
a dcpo $P$ is a continuous domain (resp., a hypercontinuous domain) if and
only if $P$ is a $C$-space with respect to the Scott topology (resp., upper
topology), we introduce the concept of _strongly continuous domains_. That is
to say, a dcpo $P$ is called a strongly continuous domain if $P$ is a
$C$-space with respect to the strong Scott topology. In this paper, we will
investigate the properties of strongly continuous domains from the aspects of
order structure and topology structure. In particular, we introduce the notion
of _strong way-below relation_ and obtain a characterization of strongly
continuous domains via this relation. We prove that the strong way-below
relation on a strongly continuous domain satisfies the interpolation property
and clarify the relationship between strongly continuous domains and
continuous domains, and the relationship between strongly continuous domains
and hypercontinuous domains. We prove that a strongly continuous domain $P$
endowed with the strong Scott topology is a locally compact sober space, and
$P$ is a $T_{2}$ space with respect to the _strong Lawson topology_ which is
the common refinement of the strong Scott topology and the lower topology.

##  2 Preliminaries

We now recall some basic definitions and notations needed in this paper. For
further details, we refer the reader to [2, 3]. For a set $X$, the family of
all finite sets in $X$ is denoted by $X^{(<\omega)}$. For a poset $L$, if $L$
has a greatest element (resp., smallest element), it is called the unit or top
element (resp., zero or bottom element) of $L$ and is written as $1$ (resp.,
$0$). For any $x\in L$ and $A\subseteq L$, let $\downarrow x=\\{y\in L:y\leq
x\\}$, $\downarrow\\!A=\bigcup\\{\downarrow\\!x:x\in A\\}$; $\uparrow\\!x$ and
$\uparrow\\!A$ are defined dually. A subset $A$ is called a _lower set_
(resp., an _upper set_) if $A=\ \downarrow\\!A$ (resp., $A=\ \uparrow\\!A$). A
nonempty subset $D$ of $L$ is called _directed_ if every two elements in $D$
have an upper bound in $D$. $L$ is called a _directed complete poset_ , or
_dcpo_ for short, if every directed subset of $L$ has the least upper bound in
$L$. A subset $I\subseteq L$ is called an _ideal_ of $L$ if $I$ is a directed
and a lower set. Dually, we define the concept of _filters_. $L$ is called an
(inf) _semilattice_ if any two elements $a,b$ have an inf, denoted by $a\wedge
b$. Dually, $L$ is a _sup semilattice_ if any two elements $a,b$ have a sup,
denoted by $a\vee b$. $L$ is called a _complete semilattice_ if $L$ is a dcpo
and every nonempty subset of $L$ has an inf. $L$ is called a _complete
lattice_ if every subset of $L$ has a sup and an inf. The _upper topology_ on
$L$, generated by $\\{L\\}\cup\\{L\setminus\\!\downarrow\\!x:x\in L\\}$ (as a
subbase), is denoted by $\upsilon(L)$. Dually, we define the _lower topology_
on $L$ and denote it by $\omega(L)$. A subset $U$ of $L$ is called _Scott
open_ if $U$ is an upper set and for any directed subset $D$ for which
$\bigvee D$ exists, $\bigvee D\in U$ implies $D\cap U\neq\emptyset$. All Scott
open subsets of $L$ form a topology, and we call this topology the _Scott
topology_ on $L$ and denote it by $\sigma(L)$. The complement of a Scott open
set is called _Scott closed_.

For a topological space $X$, let $\mathcal{O}(X)$ (resp., $\mathcal{C}(X)$) be
the lattice of all open subsets (closed subsets) in $X$. For $A\subseteq X$,
the _closure_ (resp., _interior_) of $A$ in $X$ is denoted by cl${}_{X}A$
(resp., int${}_{X}A$). For a $T_{0}$ space $X$, we use $\leq_{X}$ to represent
the _specialization order_ of $X$, that is , $x\leq_{X}y$ if and only if
$x\in{\rm cl}_{X}(\\{y\\})$. Unless otherwise stated, throughout the paper,
whenever an order-theoretic concept is mentioned, it is to be interpreted with
respect to the specialization order.

######  Definition 2.1.

([2]) Let $L$ be a poset.

  1. (1)

We say that $x$ is _way below_ $y$, in symbols $x\ll y$, iff for any directed
subset $D\subseteq L$ for which $\sup D$ exists, $y\leq\sup D$ implies the
existence of a $d\in D$ with $x\leq d$.

  2. (2)

$L$ is called _continuous_ , if for any $x\in L,\Downarrow x=\\{u\in L:u\ll
x\\}$ is directed and $x=\bigvee\Downarrow x$. A dcpo which is continuous is
called a _continuous domain_ , or _domain_. A domain which is a complete
lattice is called a _continuous lattice_.

  3. (3)

We say that $x\prec y$ iff $y\in$ int${}_{\upsilon(L)}\\!\uparrow\\!x$. And
$L$ is called _hypercontinuous_ if for any $x\in L,\\{y\in L:y\prec x\\}$ is
directed and $x=\bigvee\\{y\in L:y\prec x\\}$. A dcpo which is hypercontinuous
is called a _hypercontinuous domain_. A hypercontinuous domain which is a
complete lattice is called a _hypercontinuous lattice_.

######  Definition 2.2.

([4]) Let $L$ be a complete lattice. A binary relation $\lhd$ on $L$ is
defined as follows: $x\lhd y$ iff for any $S\subseteq L,y\leq\bigvee S$
implies that $x\in\ \\!\downarrow\\!S$. $L$ is a _prime continuous lattice_ if
for all $x\in L,x=\bigvee\\{y\in L:y\lhd x\\}$.

######  Definition 2.3.

([14]) Let $X$ be a $T_{0}$ space. $X$ is called a $C$-space if for any
$U\in\mathcal{O}(X)$ and $x\in U$, there exists $y\in U$ such that $x\in\\!$
int${}_{X}\\!\uparrow\\!y\subseteq\ \\!\uparrow\\!y\subseteq U$.

######  Proposition 2.4.

([14]) For a $T_{0}$ space $X$, $X$ is a $C$-space if and only if
$(\mathcal{O}(X),\subseteq)$ is a completely distributive lattice.

######  Theorem 2.5.

([2]) Let $L$ be a dcpo.

  1. (1)

$L$ is a domain iff $(L,\sigma(L))$ is a $C$-space.

  2. (2)

$L$ is a hypercontinuous domain iff $(L,\upsilon(L))$ is a $C$-space.

######  Theorem 2.6.

([2]) Let $L$ be a complete lattice. The following conditions are equivalent:

  1. (1)

$L$ is completely distributive;

  2. (2)

$L$ is distributive, and both $L$ and $L^{\rm{op}}$ are continuous lattices;

  3. (3)

$L$ is continuous and every element is the sup of co-primes.

######  Definition 2.7.

([8]) Let $L$ be a dcpo. A subset $U\subseteq L$ is called _strongly Scott
open_ if (i) $U=\ \uparrow\\!U$, and (ii) for any directed subset $D$ of $L$
and $x\in L$, $\bigcap_{d\in D}\uparrow\\!d~{}\cap\uparrow\\!x\subseteq U$
(that is $\uparrow\bigvee D\ \cap\uparrow x\subseteq U$) implies
$\uparrow\\!d\ \cap\uparrow\\!x\subseteq U$ for some $d\in D$. Let
$\sigma^{s}(L)$ denote the set of all strongly Scott open subsets of $L$.

Clearly, if $U,V\in\sigma^{s}(L)$, then $U\cap V\in\sigma^{s}(L)$. The
topology generated by $\sigma^{s}(L)$ (as a base) is called the _strong Scott
topology_ on $L$ and denote it by $\sigma_{s}(L)$. The space
$(L,\sigma_{s}(L))$ is called the _strong Scott space_ of $L$, and will be
denoted by $\sum_{s}L$.

In order not to cause ambiguity, the elements in $\sigma_{s}(L)$ will be
called _strong Scott topology open sets_.

######  Proposition 2.8.

([8]) Let $L$ be a poset, then
$\upsilon(L)\subseteq\sigma^{s}(L)\subseteq\sigma_{s}(L)\subseteq\sigma(L)$.

##  3 Strongly continuous domains

In this section we introduce the notion of strongly continuous domains and
give some characterizations of strongly continuous domains from the aspect of
order structure.

######  Definition 3.1.

Let $L$ be a dcpo. $L$ is called a _strongly continuous domain_ if
$(L,\sigma_{s}(L))$ is a $C$-space, i.e., for any $U\in\sigma_{s}(L)$ and
$x\in L$, $x\in U$ implies there exists a $u\in U$ such that
$x\in{\rm{int}}_{\sigma_{s}(L)}\\!\uparrow\\!u\subseteq\
\\!\uparrow\\!u\subseteq U$.

The reason we adopt the terminology “strongly continuous” in the paper is that
we introduce the notion of strongly continuous domains via strong Scott
topology. It should be pointed out that the terminology “strongly continuous”
in this paper is completely different from that in [5].

######  Definition 3.2.

([5]) Let $P$ be a poset and $x,y\in P$. We write $x\ll_{l}y$, if for any
directed set $D$ and any upper bound $z$ of $D$ with $y\leq\sup_{z}D$, there
is a $d\in D$ such that $x\leq d$, where $\sup_{z}D$ denotes the supremum of
$D$ in $\downarrow\\!z$. $P$ is called a strongly continuous poset if for all
$x\in P$, $\Downarrow_{l}x=\\{y\in L:y\ll_{l}x\\}$ is directed and
$\sup\Downarrow_{l}x=x$.

######  Definition 3.3.

Let $L$ be a poset. A binary relation $\ll_{s}$ on $L$ is defined as follows:
$x\ll_{s}y$ if for any directed subset $D\subseteq L$ and $a\in L$,
$\bigcap_{d\in D}\\!\uparrow\\!d\ \cap\uparrow\\!a\subseteq\ \\!\uparrow\\!y$
implies $\uparrow\\!d\ \cap\uparrow\\!a\subseteq\ \\!\uparrow\\!x$ for some
$d\in D$.

######  Remark 3.4.

Let $L$ be a dcpo and $u,x,y,z\in L$. We have the following.

  1. (1)

$x\ll_{s}y$ implies $x\ll y$, and $x\ll_{s}y$ is equivalent to $x\ll y$ if $L$
is a sup semilattice.

  2. (2)

$u\leq x\ll_{s}y\leq z$ implies $u\ll_{s}z$.

  3. (3)

$x\ll_{s}z$, $y\ll_{s}z$ implies $x\vee y\ll_{s}z$ whenever $x\vee y$ exists
in $L$.

  4. (4)

$0\ll_{s}x$ whenever $L$ has a smallest element $0$.

Hence $\ll_{s}$ is an auxiliary relation on $L$.

For $x\in L$, we write $\Downarrow_{s}x=\\{y\in L:y\ll_{s}x\\}$ and
$\Uparrow_{s}x=\\{y\in L:x\ll_{s}y\\}$.

######  Proposition 3.5.

Let $L$ be a dcpo. Suppose that there exists a directed set $D\subseteq\
\Downarrow_{s}\\!x$ such that $\sup D=x$, then we have the following
conditions.

  1. (1)

$\Downarrow_{s}\\!x$ is directed and $\sup\Downarrow_{s}\\!x=x$;

  2. (2)

If $y\ll_{s}x$ in $\downarrow\\!x$, then $y\ll_{s}x$ in $L$.

######  Proof 3.6.

(1): Let $y_{1},y_{2}\in\ \\!\Downarrow_{s}\\!\\!x$, i.e., $y_{1}\ll_{s}x$ and
$y_{2}\ll_{s}x$. Since $\ll_{s}\ \\!\subseteq\ \\!\ll$, then $y_{1}\ll x$ and
$y_{2}\ll x$. Since $\textrm{sup}D=x$, then $y_{1}\leq d_{1},y_{2}\leq d_{2}$
for some $d_{1},d_{2}\in D$. Since $D$ is directed, there exists $d_{3}\in D$
such that $y_{1}\leq d_{1}\leq d_{3}$, $y_{2}\leq d_{2}\leq d_{3}$, then
$d_{3}\geq y_{1},y_{2}$. Thus $\Downarrow_{s}\\!x$ is directed by $d_{3}\in
D\subseteq\ \\!\Downarrow_{s}\\!x$. Since
$x=\textrm{sup}D\leq\textrm{sup}\Downarrow_{s}\\!x\leq\textrm{sup}\downarrow\\!x=x$,
we have $\textrm{sup}\Downarrow_{s}\\!x=x$.
(2): Let $y\ll_{s}x$ in $\downarrow\\!x$. Then $y\ll x$. Since directed set
$D\subseteq\ \\!\Downarrow_{s}\\!x$ and $\textrm{sup}D=x$, $y\leq d$ for some
$d\in D$. Therefore $y\ll_{s}x$ by $d\in\ \\!\Downarrow_{s}\\!x$.

######  Theorem 3.7.

Let $L$ be a dcpo. The following two conditions are equivalent:

  1. (1)

$L$ is a strongly continuous domain;

  2. (2)

For any $x\in L$, $\Downarrow_{s}\\!x$ is directed, $x=\sup\Downarrow_{s}\\!x$
and $\Uparrow_{s}\\!x\in\sigma_{s}(L)$.

######  Proof 3.8.

$(1)\Rightarrow(2)$: Suppose that $L$ is a strongly continuous domain. Then
$(L,\sigma_{s}(L))$ is a $C$-space. For any $a\in L$, let $H_{a}=\\{x\in
L:a\in\textrm{int}_{\sigma_{s}(L)}(\uparrow\\!\\!x)\\}$. Then since
$(L,\sigma_{s}(L))$ is a $C$-space and $a\in L\in\sigma_{s}(L)$, there exists
$x\in L$ such that
$a\in\textrm{int}_{\sigma_{s}(L)}(\uparrow\\!\\!x)\subseteq\
\\!\uparrow\\!\\!x\subseteq L$. Hence $H_{a}\neq\emptyset$. For any
$x_{1},x_{2}\in H_{a}$, since $(L,\sigma_{s}(L))$ is a $C$-space, there exists
$x_{3}\in L$ such that
$a\in\textrm{int}_{\sigma_{s}(L)}(\uparrow\\!\\!x_{3})\subseteq\
\\!\uparrow\\!\\!x_{3}\subseteq(\textrm{int}_{\sigma_{s}(L)}(\uparrow\\!\\!x_{1}))\cap(\textrm{int}_{\sigma_{s}(L)}(\uparrow\\!\\!x_{2}))$,
hence $x_{3}\in H_{a}$ and $x_{1},x_{2}\leq x_{3}$, thus $H_{a}$ is directed.
If $x\in H_{a}$ and $y\leq x$, then
$a\in\textrm{int}_{\sigma_{s}(L)}(\uparrow\\!\\!x)\subseteq\textrm{int}_{\sigma_{s}(L)}(\uparrow\\!\\!y)$,
thus $y\in H_{a}$, hence $H_{a}$ is a lower set. We now show that $\bigvee
H_{a}=a$. Obviously, $a$ is an upper bound of $H_{a}$. Let $b$ be an other
upper bound of $H_{a}$, i.e., $H_{a}\subseteq\ \\!\downarrow\\!b$. Assume that
$a\nleq b$, then $a\in
L\setminus\downarrow\\!b\in\upsilon(L)\subseteq\sigma_{s}(L)$, since
$(L,\sigma_{s}(L))$ is a $C$-space, there exists $x\in
L\setminus\downarrow\\!b$ such that
$a\in\textrm{int}_{\sigma_{s}(L)}(\uparrow\\!\\!x)\subseteq\
\\!\uparrow\\!\\!x\subseteq L\setminus\downarrow\\!b$. Hence $x\in H_{a}$ and
$x\nleq b$, this is a contradiction. Thus $\bigvee H_{a}=a$. We now show that
$H_{a}=\ \\!\Downarrow_{s}\\!a$. On the one hand, if $x\in H_{a}$, then there
exists $U\in\sigma^{s}(L)$ such that $a\in
U\subseteq\textrm{int}_{\sigma_{s}(L)}(\uparrow\\!\\!x)$. For any directed
subset $D\subseteq L$ and $z\in L$, if $\uparrow\\!\bigvee
D~{}\cap\uparrow\\!z\subseteq\ \\!\uparrow\\!a$, then $\uparrow\\!\bigvee D\
\\!\cap\uparrow\\!z\subseteq\ \\!\uparrow\\!a\subseteq U\in\sigma^{s}(L)$,
hence $\uparrow\\!d~{}\cap\uparrow\\!z\subseteq
U\subseteq\textrm{int}_{\sigma_{s}(L)}(\uparrow\\!\\!x)\subseteq\
\\!\uparrow\\!x$ for some $d\in D$. Thus $x\ll_{s}a$. Therefore
$H_{a}\subseteq\ \\!\Downarrow_{s}\\!a$. On the other hand, for any $y\in\
\\!\Downarrow_{s}\\!a$, i.e., $y\ll_{s}a=\bigvee H_{a}$, then $y\ll a=\bigvee
H_{a}$. Since $H_{a}$ is an ideal, $y\in\ \\!\downarrow\\!H_{a}=H_{a}$. Now we
prove that $\Uparrow_{s}\\!\\!a\in\sigma_{s}(L)$. Since $x\in\
\\!\Uparrow_{s}\\!\\!a$ iff $a\in\ \\!\Downarrow_{s}\\!x=H_{x}$ iff
$x\in\textrm{int}_{\sigma_{s}(L)}(\uparrow\\!\\!a)$, we have
$\Uparrow_{s}\\!a=\textrm{int}_{\sigma_{s}(L)}(\uparrow\\!a)\in\sigma_{s}(L)$.
$(2)\Rightarrow(1)$: Suppose $a,b\in L$ such that $a\ll_{s}b$. Since
$\Uparrow_{s}\\!\\!a\in\sigma_{s}(L)$ and $\Uparrow_{s}\\!\\!a\subseteq\
\\!\uparrow\\!\\!a$, then we have $b\in\
\\!\Uparrow_{s}\\!a\subseteq\textrm{int}_{\sigma_{s}(L)}(\uparrow\\!\\!a)$. We
now show that $(L,\sigma_{s}(L))$ is a $C$-space. Suppose that
$U\in\sigma_{s}(L)$ and $b\in U$, then $\bigvee\\{x\in L:x\ll_{s}b\\}=b\in U$.
Since $\sigma_{s}(L)\subseteq\sigma(L)$, $\bigvee\\{x\in L:x\ll_{s}b\\}=b\in
U\in\sigma_{s}(L)\subseteq\sigma(L)$, thus $\\{x\in L:x\ll_{s}b\\}\cap
U\neq\emptyset$, hence $x\in U$ for some $x\in\ \\!\Downarrow_{s}\\!b$. Thus
$b\in\textrm{int}_{\sigma_{s}(L)}(\uparrow\\!x)$. Therefore
$(L,\sigma_{s}(L))$ is a $C$-space.

According to the Definition 3.1 and Theorem 3.7, strongly continuous domains
can be characterized by means of strong way-below relation.

######  Proposition 3.9.

Let $L$ be a strongly continuous domain. If $x\ll_{s}z$ and $z\leq\bigvee D$
for a directed set in $L$, then $x\ll_{s}d$ for some $d\in D$.

######  Proof 3.10.

Let $I=\bigcup\\{\Downarrow_{s}\\!d:d\in D\\}$. Since $L$ is strongly
continuous, $\bigvee I=\bigvee D$ and $I$ is an ideal. If $x\ll_{s}z$ and
$z\leq\bigvee D$, then $x\ll z\leq\bigvee D=\bigvee I$. Since strongly
continuous domains are continuous domains by definition, $x\in I$. Thus
$x\ll_{s}d$ for some $d\in D$.

######  Proposition 3.11.

In a strongly continuous domain $L$, $\ll_{s}$ satisfies the interpolation
property, i.e., for all $x,y\in L$, $x\ll_{s}y$ implies that there exists a
$z\in L$ such that $x\ll_{s}z\ll_{s}y$.

######  Proof 3.12.

Suppose $x,y\in L$ such that $x\ll_{s}y$, we now show that $x\ll_{s}z\ll_{s}y$
for some $z\in L$. Let
$D=\bigcup_{a\in\Downarrow_{s}y}\Downarrow_{s}\\!a=\\{b\in L:$ there is an
$a\in L$ with $b\ll_{s}a\ll_{s}y\\}$. We claim that $D$ is directed. For any
$b_{1},b_{2}\in D$, we have $b_{1}\ll_{s}a_{1}\ll_{s}y$ for some $a_{1}\in L$
and $b_{2}\ll_{s}a_{2}\ll_{s}y$ for some $a_{2}\in L$. Since
$\Downarrow_{s}\\!y$ is directed, there exists $a_{3}\in\
\\!\Downarrow_{s}\\!y$ such that $a_{1},a_{2}\leq a_{3}$. Hence
$b_{1},b_{2}\ll_{s}a_{3}$. Since $\Downarrow_{s}\\!a_{3}$ is directed, there
exists $b_{3}\in\ \\!\Downarrow_{s}\\!a_{3}$ such that $b_{1},b_{2}\leq
b_{3}\ll_{s}a_{3}\ll_{s}y$. Thus $b_{3}\in\\!D$ and $b_{1},b_{2}\leq b_{3}$.
Therefore $D$ is directed. Since $\bigvee
D=\bigvee\\{\bigvee\Downarrow_{s}\\!a:a\in\
\\!\Downarrow_{s}\\!y\\}=\bigvee\\{a:a\in\
\\!\Downarrow_{s}\\!y\\}=\bigvee\Downarrow_{s}\\!y=y$, $\bigcap_{d\in
D}\\!\uparrow\\!d\subseteq\ \\!\uparrow\\!y$. It follows from $x\ll_{s}y$ that
$\uparrow\\!d\subseteq\ \\!\uparrow\\!x$ for some $d\in\\!D$. Thus there
exists $a\in L$ such that $d\ll_{s}a\ll_{s}y$. Therefore $x\ll_{s}a\ll_{s}y$.

Recall that hypercontinuous domains are continuous domains. Now we discuss the
relationship between strongly continuous domains and hypercontinuous domains,
and the relationship between strongly continuous domains and continuous
domains.

######  Lemma 3.13.

([2]) Let $L$ be a dcpo. Then the following conditions are equivalent:

  1. (1)

$L$ is hypercontinuous;

  2. (2)

$L$ is continuous and $\ll=\prec$;

  3. (3)

$L$ is continuous and $\upsilon(L)=\sigma(L)$.

######  Theorem 3.14.

Let $L$ be a dcpo. Then the following conditions are equivalent:

  1. (1)

$L$ is hypercontinuous;

  2. (2)

$L$ is strongly continuous and $\ll_{s}=\prec$;

  3. (3)

$L$ is strongly continuous and $\upsilon(L)=\sigma_{s}(L)$.

######  Proof 3.15.

$(1)\Rightarrow(2)$: Let $L$ be a hypercontinuous domain, thus $\ll=\prec$ by
Lemma 3.13. Since $\prec\subseteq\ll_{s}\subseteq\ll$, we have
$\prec=\ll_{s}$. For any $x\in L$, $\Uparrow_{s}\\!x=\\{y\in L:x\prec
y\\}=\textrm{int}_{\upsilon(L)}\\!\uparrow\\!x\in\upsilon(L)\subseteq\sigma_{s}(L)$.
$(2)\Rightarrow(3)$: Obviously, $\upsilon(L)\subseteq\sigma_{s}(L)$. For each
$U\in\sigma_{s}(L)$, we show that $U\in\upsilon(L)$. For any $x\in U$,
$x=\bigvee^{\uparrow}\\!\Downarrow_{s}\\!x=\bigvee^{\uparrow}\\{y\in L:y\prec
x\\}\in U\in\sigma_{s}(L)\subseteq\sigma(L)$, hence there is a $y\prec x$ with
$y\in U$. Since $y\prec x$, $x\in\textrm{int}_{\upsilon(L)}\\!\uparrow\\!y$.
Then $x\in\textrm{int}_{\upsilon(L)}\\!\uparrow\\!y\subseteq\
\\!\uparrow\\!y\subseteq U$. Thus $U\in\upsilon(L)$. Therefore
$\upsilon(L)=\sigma_{s}(L)$.
$(3)\Rightarrow(1)$: $L$ is strongly continuous iff $(L,\sigma_{s}(L))$ is a
$C$-space iff $(L,\upsilon(L))$ is a $C$-space iff $L$ is hypercontinuous.

######  Example 3.16.

Let $L=\\{0\\}\cup\\{a_{1},a_{2},\cdots,a_{n},\cdots\\}$ with the order
generated by

  1. (a)

$a_{n}>0$ for all $n\in\mathbb{N}$;

  2. (b)

There is no order relationship between $a_{i}$ and $a_{j}$ for all
$i,j\in\mathbb{N}$ (Fig. 1).

Then $L$ is a strongly continuous domain, but not a hypercontinuous domain.

Figure 1: The poset $L$.

######  Proof 3.17.

Since the directed subset in $L$ can only be a single point set or a binary
set $\\{0,a_{n}\\}(n\in\mathbb{N})$, $L$ is a dcpo. It is easy to show that
$L$ is not a hypercontinous domain. Now we prove that $L$ is a strongly
continuous domain. Let $x\in L$, we now show that $\Downarrow_{s}\\!x$ is
directed. Obviously, $0\ll_{s}0$, then $\Downarrow_{s}\\!0=\\{0\\}$. For each
$n\in\mathbb{N}$, let directed set $D\subseteq L$ and $z\in L$ such that
$\bigcap_{d\in D}\uparrow\\!d\ \\!\cap\uparrow\\!z\subseteq\
\\!\uparrow\\!a_{n}=\\{a_{n}\\}$. Then there are only the following cases: (i)
$D=\\{0\\}$, $z=\\{a_{n}\\}$; (ii) $D=\\{a_{n}\\}$, $z\in L$; (iii)
$D=\\{a_{m}\\}(m\neq n)$, $z\neq\\{0\\}$ and $z\neq\\{a_{m}\\}$; (iv)
$D=\\{0,a_{n}\\}$, $z\in L$; (v) $D=\\{0,a_{m}\\}(m\neq n)$, $z\neq\\{0\\}$
and $z\neq\\{a_{m}\\}$. Clearly, there exists $d\in D$ such that
$\uparrow\\!d\ \\!\cap\uparrow\\!z\subseteq\ \\!\uparrow\\!a_{n}=\\{a_{n}\\}$
in each case. Hence $a_{n}\ll_{s}a_{n}$. therefore
$\Downarrow_{s}\\!a_{n}=\\{0,a_{n}\\}$. Thus $\Downarrow_{s}\\!x$ is directed
and $x=\bigvee\\!\Downarrow_{s}\\!x$ for all $x\in L$. Now we prove that
$\Uparrow_{s}\\!x\in\sigma_{s}(L)$. Clearly,
$\Uparrow_{s}\\!0=L\in\sigma_{s}(L)$. For each $n\in\mathbb{N}$,
$\Uparrow_{s}\\!a_{n}=\\{a_{n}\\}$ and
$\Uparrow_{s}\\!a_{n}\in\sigma^{s}(L)\subseteq\sigma_{s}(L)$ according to the
above proof. Therefore $L$ is a strongly continuous domain.

According to the Theorem 3.14 and Example 3.16, a hypercontinuous domain is a
strongly continuous domain, but the converse is not true.

######  Theorem 3.18.

Let $L$ be a dcpo. Then the following conditions are equivalent:

  1. (1)

$L$ is strongly continuous;

  2. (2)

$L$ is continuous, $\ll_{s}=\ll$ and $\Uparrow_{s}\\!x\in\sigma_{s}(L)$ for
all $x\in L$;

  3. (3)

$L$ is continuous and $\sigma_{s}(L)=\sigma(L)$.

######  Proof 3.19.

$(1)\Rightarrow(2)$: Since $L$ is strongly continuous and
$\ll_{s}\subseteq\ll$, $L$ is continuous. If $x\ll y=\bigvee^{\uparrow}\\{z\in
L:z\ll_{s}y\\}$, then there exists $z\in\ \\!\Downarrow_{s}\\!y$ such that
$x\leq z$, hence $x\ll_{s}y$. Clearly, $\Uparrow_{s}\\!x\in\sigma_{s}(L)$ for
all $x\in L$.
$(2)\Rightarrow(3)$: We only need prove that
$\sigma(L)\subseteq\sigma_{s}(L)$. Let $U\in\sigma(L)$, for each $x\in U$,
$x=\bigvee^{\uparrow}\\!\Downarrow\\!x=\bigvee^{\uparrow}\\!\Downarrow_{s}\\!x\in
U\in\sigma(L)$, then there exists $y\in\ \\!\Downarrow_{s}\\!x$ such that
$y\in U$. Thus $x\in\ \\!\Uparrow_{s}\\!y\subseteq\ \\!\uparrow\\!y\subseteq
U$. Since $\Uparrow_{s}\\!y\in\sigma_{s}(L)$, $U\in\sigma_{s}(L)$.
$(3)\Rightarrow(1)$: $L$ is continuous iff $(L,\sigma(L))$ is a $C$-space iff
$(L,\sigma_{s}(L))$ is a $C$-space iff $L$ is strongly continuous.

The following example shows that a continuous domain need not be a strongly
continuous domain.

######  Example 3.20.

([8]) Let $C=\\{a_{1},a_{2},\cdots,a_{n},\cdots\\}\cup\\{\omega_{0}\\}$ and
$L=C\cup\\{b\\}\cup\\{\omega_{1},\omega_{2},\cdots,\omega_{n},\cdots\\}$ with
the order generated by

  1. (a)

$a_{1}<a_{2}<\cdots<a_{n}<a_{n+1}<\cdots$;

  2. (b)

$a_{n}<\omega_{0}$ for all $n\in\mathbb{N}$;

  3. (c)

$b<\omega_{n}$ and $a_{m}<\omega_{n}$ for all $n,m\in\mathbb{N}$ with $m\leq
n$ (Fig. 2).

Figure 2: The poset $L$.

Then $L$ is a dcpo and $D$ is a directed set of $L$ iff $D\subseteq C$ or $D$
has a largest element, and hence $x\ll x$ for all $x\in
L\setminus\\{\omega_{0}\\}$. Therefore $L$ is a continuous domain. However,
$L$ is not a strongly continuous domain. In fact, choose
$D=\\{a_{n}:n\in\mathbb{N}\\}$, then $\bigcap_{d\in D}\\!\uparrow\\!\\!d\
\\!\cap\\!\uparrow\\!\\!b\subseteq\
\\!\uparrow\\!\\!\omega_{1}=\\{\omega_{1}\\}$, but
$\uparrow\\!\\!a_{n}~{}\cap\uparrow\\!\\!b=\\{\omega_{n},\omega_{n+1},\cdots\\}\nsubseteq\
\\!\uparrow\\!\\!\omega_{1}$ for each $n\in\mathbb{N}$. Hence
$\omega_{1}\notin\ \\!\Downarrow_{s}\\!\\!\omega_{1}$. Since
$\Downarrow_{s}\\!\omega_{1}\subseteq\
\\!\downarrow\\!\omega_{1}=\\{\omega_{1},a_{1},b\\}$,
$\Downarrow_{s}\\!\omega_{1}\subseteq\\{a_{1},b\\}$, we have
$\omega_{1}\neq\bigvee\Downarrow_{s}\\!\omega_{1}$.

According to the Theorem 3.18 and Example 3.20, a strongly continuous domain
is a continuous domain, but the converse is not true. Therefore strongly
continuous domains lie strictly between hypercontinuous domains and continuous
domains.

##  4 Topologies on strongly continuous domains

In this section, we study some properties of strongly continuous domains
endowed with strong Scott topology and strong Lawson topology.

######  Proposition 4.1.

Let $L$ be a dcpo. Then we have the following conditions.

  1. (1)

$\textrm{cl}_{\sigma_{s}(L)}\\{x\\}=\ \downarrow\\!x$ for all $x\in L$;

  2. (2)

$\sigma_{s}(L)$ is a $T_{0}$-topology;

  3. (3)

If $A=\ \uparrow\\!A$, then $A=\bigcap\\{U\in\sigma_{s}(L):A\subseteq U\\}$.

######  Proposition 4.2.

Let $L$ be a dcpo. Consider the following two conditions:

  1. (1)

$y\in\textrm{int}_{\sigma_{s}(L)}\uparrow\\!x$;

  2. (2)

$x\ll_{s}y$.

Then $(1)\Rightarrow(2)$; if $L$ is strongly continuous, then
$(2)\Rightarrow(1)$, and $(1)$ and $(2)$ are equivalent.

######  Proof 4.3.

$(1)\Rightarrow(2)$: By the proof of Theorem 3.7.
$(2)\Rightarrow(1)$: Let $x\ll_{s}y$, then $y\in\
\\!\Uparrow_{s}\\!\\!x\subseteq\ \\!\uparrow\\!\\!x$. Since $L$ is strongly
continuous, $\Uparrow_{s}\\!x\in\sigma_{s}(L)$. Hence $y\in\
\\!\Uparrow_{s}\\!x\subseteq\textrm{int}_{\sigma_{s}(L)}\uparrow\\!x$.

######  Proposition 4.4.

Let $L$ be a strongly continuous domain. Then we have the following
conditions.

  1. (1)

If $U=\ \\!\uparrow\\!U$, then $U\in\sigma_{s}(L)$ iff there is a $u\in U$
such that $x\in\ \\!\Uparrow_{s}\\!u$ for all $x\in U$;

  2. (2)

$\\{\Uparrow_{s}\\!u:u\in L\\}$ form a basis for the strong Scott topology;

  3. (3)

$\textrm{int}_{\sigma_{s}(L)}\uparrow\\!x=\ \\!\Uparrow_{s}\\!x$ for all $x\in
L$;

  4. (4)

For any subset $X\subseteq L$,
$\textrm{int}_{\sigma_{s}(L)}X=\bigcup\\{\Uparrow_{s}\\!u:\Uparrow_{s}\\!u\subseteq
X\\}$.

######  Proof 4.5.

(1) For each $x\in U$, since $L$ is strongly continuous,
$\Downarrow_{s}\\!\\!x$ is directed and $x=\bigvee\Downarrow_{s}\\!x$. And
hence $x=\bigvee\Downarrow_{s}\\!x\in\sigma_{s}(L)\subseteq\sigma(L)$, thus
$\Downarrow_{s}\\!x\cap U\neq\emptyset$. Therefore there exists $u\in U$ such
that $x\in\ \\!\Uparrow_{s}\\!u$. On the other hand, it is trivial.
(2) It follows directly from (1).
(3) It follows from Proposition 4.2.
(4) On the one hand, obviously, $\bigcup\\{\Uparrow_{s}\\!\\!u:\
\\!\Uparrow_{s}\\!\\!u\subseteq X\\}\subseteq X$ and
$\bigcup\\{\Uparrow_{s}\\!\\!u:\ \\!\Uparrow_{s}\\!\\!u\subseteq
X\\}\in\sigma_{s}(L)$, then $\bigcup\\{\Uparrow_{s}\\!\\!u:\
\\!\Uparrow_{s}\\!\\!u\subseteq X\\}\subseteq\textrm{int}_{\sigma_{s}(L)}X$.
On the other hand, for each $y\in\textrm{int}_{\sigma_{s}(L)}X$, there exists
$u\in\textrm{int}_{\sigma_{s}(L)}X$ such that $y\in\
\\!\Uparrow_{s}\\!u\subseteq\textrm{int}_{\sigma_{s}(L)}X$. Thus
$\Uparrow_{s}\\!u=\textrm{int}_{\sigma_{s}(L)}\uparrow\\!u\subseteq\
\\!\uparrow\\!u\subseteq\textrm{int}_{\sigma_{s}(L)}X\subseteq X$. Therefore
$\textrm{int}_{\sigma_{s}(L)}X\subseteq\bigcup\\{\Uparrow_{s}\\!u:\Uparrow_{s}\\!u\subseteq
X\\}$.

Now we recall the definitions of _prime elements_ , _sober spaces_. An element
$p$ in a poset $L$ is called _prime_ iff $p=1$ or $L\setminus\downarrow p$ is
a filter. An element is _co-prime_ iff it is prime of $L^{\textrm{op}}$. The
sets of all prime and co-prime elements of $L$ are denoted by
$\textrm{PRIME}(L)$ and $\textrm{COPRIME}(L)$, respectively. A nonempty subset
$A$ of a $T_{0}$ space $X$ is said to be _irreducible_ if for any
$\\{B,C\\}\subseteq\mathcal{C}(X)$, $A\subseteq B\cup C$ implies $A\subseteq
B$ or $A\subseteq C$. Obviously, $\textrm{cl}\\{x\\}$ is a irreducible closed
set for all $x\in X$. A space is called _sober_ , if for any irreducible
closed set $C$, there is a unique point $x\in X$ such that
$C=\textrm{cl}\\{x\\}$.

######  Lemma 4.6.

([2]) Let $X$ be a space and $A$ a subset of $X$. Then the following
conditions are equivalent:

  1. (1)

$A$ is a irreducible closed set of $X$;

  2. (2)

$A$ is a co-prime in lattice $\mathcal{C}(X)$;

  3. (3)

$X\setminus A$ is a prime in lattice $\mathcal{O}(X)$;

  4. (4)

For any $U,V\in\mathcal{O}(X)$, if $U\cap A\neq\emptyset$ and $V\cap
A\neq\emptyset$, then $U\cap V\cap A\neq\emptyset$.

######  Definition 4.7.

Let $L$ be a dcpo. The set of all strong Scott topology open filters of $L$ is
denoted by $\textrm{SOFilt}(L)$, i.e., $U\in\textrm{SOFilt}(L)$ iff
$U\in\sigma_{s}(L)$ and $U$ is a filter.

######  Proposition 4.8.

Let $L$ be a dcpo and $U\in\sigma_{s}(L)$.

  1. (1)

$U$ is a co-prime in $\sigma_{s}(L)$ iff $U\in\textrm{SOFilt}(L)$;

  2. (2)

$L\setminus\downarrow\\!a\in\textrm{PRIME}(\sigma_{s}(L))$ for all $a\in L$;
For each $U\in\textrm{PRIME}(\sigma_{s}(L)),U\neq L$, if $L$ is strongly
continuous, then there exists $a\in L$ such that $U=L\setminus\downarrow\\!a$.

######  Proof 4.9.

(1): On the one hand, suppose that $U\in\textrm{COPRIME}(\sigma_{s}(L))$, we
only need to show that $U$ is a filter. For any $x,y\in U$, assume that $U\
\\!\cap\downarrow\\!x\ \\!\cap\downarrow\\!y=\emptyset$, then
$U\subseteq(L\setminus\downarrow x)\cup(L\setminus\downarrow y)$. Since
$U\in\textrm{COPRIME}(\sigma_{s}(L))$, $U\subseteq L\setminus\downarrow x$ or
$U\subseteq L\setminus\downarrow y$, which is in contradiction with $U$ is an
upper set. Therefore there is a $z\in U$ such that $z\leq x$ and $z\leq y$. On
the other hand, suppose that $U\in\textrm{SOFilt}(L)$, if
$V,W\in\sigma_{s}(L)$ and $U\nsubseteq V,U\nsubseteq W$, we only need check
$U\nsubseteq V\cup W$. Since $U\nsubseteq V$ and $U\nsubseteq W$, then there
exists $v\in U\setminus V$ and $w\in U\setminus W$. Note that $U$ is a filter,
hence there exists $u\in U$ such that $u\leq v,w$, then $u\in U\setminus(V\cup
W)$. Therefore $U\nsubseteq V\cup W$.
(2): By Proposition 4.1 and Lemma 4.6,
$L\setminus\downarrow\\!a=L\setminus\textrm{cl}_{\sigma_{s}(L)}\\{a\\}\in\textrm{PRIME}(\sigma_{s}(L))$
for all $a\in L$. Conversely, for each $U\in\textrm{PRIME}(\sigma_{s}(L))$ and
$U\neq L$, let $A=L\setminus U$, then $A$ is an irreducible closed set in
$(L,\sigma_{s}(L))$. We have to show that $A$ has a largest element $e$; since
$A$ is a lower set, this will show that $A=\ \\!\downarrow\\!e$ as desired.
Let $A^{\ast}=\bigcup\\{\Downarrow_{s}a:a\in A\\}=\
\\!\Downarrow_{s}\\!A\subseteq\ \\!\downarrow\\!A=A$, we claim that $A^{\ast}$
is directed. For each $b,c\in A^{\ast}$, there exists $a_{b},a_{c}\in A$ such
that $b\ll_{s}a_{b}$ and $c\ll_{s}a_{c}$, respectively. We first show that
$\Uparrow_{s}\\!b\ \\!\cap\Uparrow_{s}\\!c\cap A\neq\emptyset$, if not, then
$\Uparrow_{s}\\!b\ \\!\cap\Uparrow_{s}\\!c\subseteq U$, but
$\Uparrow_{s}\\!b,\Uparrow_{s}\\!c\in\sigma_{s}(L)$ and
$U\in\textrm{PRIME}(\sigma_{s}(L))$, then $\Uparrow_{s}\\!b\subseteq U$ or
$\Uparrow_{s}\\!c\subseteq U$. But $\Uparrow_{s}b$ contains an $a_{b}\in
A=L\setminus U$ which is impossible, similarly $\Uparrow_{s}\\!c\subseteq U$
is impossible. Pick $a\in\ \\!\Uparrow_{s}\\!b\ \\!\cap\Uparrow_{s}\\!c\cap
A$, since $\Uparrow_{s}\\!b\ \\!\cap\Uparrow_{s}\\!c\in\sigma_{s}(L)$, there
exists $d\in\ \\!\Uparrow_{s}\\!b\ \\!\cap\Uparrow_{s}\\!c$ such that $a\in\
\\!\Uparrow_{s}\\!d\subseteq\ \\!\Uparrow_{s}\\!b\ \\!\cap\Uparrow_{s}\\!c$.
Thus $d\in A^{\ast}$ and $b,c\leq d$. Hence $A^{\ast}$ is directed and
$\bigvee A^{\ast}$ exists. Note that $A^{\ast}\subseteq A$ and $A$ is a strong
Scott topology closed set, then $A$ is also a Scott closed set. Let $e=\bigvee
A^{\ast}$, then $e=\bigvee A^{\ast}\in A$. Now we show that $e$ is the largest
element in $A$. For any $x\in A$, since $\Downarrow_{s}\\!x\subseteq
A^{\ast}$, $x=\bigvee\Downarrow_{s}\\!x\leq\bigvee A^{\ast}=e$.

######  Corollary 4.10.

Let $L$ be a strongly continuous domain. Then $(L,\sigma_{s}(L))$ is a sober
space.

######  Proof 4.11.

By Lemma 4.6 and Proposition 4.8.

######  Corollary 4.12.

Let $L$ be a strongly continuous domain. Then $(L,\sigma_{s}(L))$ is a locally
compact sober space. If $L$ has a smallest element, then $(L,\sigma_{s}(L))$
is compact.

######  Proof 4.13.

Suppose that $x\in U\in\sigma_{s}(L)$, by Proposition 4.4, there exists $y\in
U$ such that $x\in\ \\!\Uparrow_{s}\\!y\subseteq\ \\!\uparrow\\!y\subseteq U$.
Note that $\uparrow\\!y$ is compact with respect to any topology whose open
sets are upper sets, thus the assertion is proved. If $L$ has a smallest
element, let $\\{U_{i}:i\in I\\}\subseteq\sigma_{s}(L)$ and
$L\subseteq\bigcup_{i\in I}U_{i}$, then there exists $i\in I$ such that $0\in
U_{i}$, hence $L=\ \\!\uparrow\\!0\subseteq U_{i}$. Therefore
$(L,\sigma_{s}(L))$ is compact.

######  Theorem 4.14.

Let $L$ be a dcpo. Then the following conditions are equivalent:

  1. (1)

$L$ is strongly continuous;

  2. (2)

For each $x\in L$, $\Uparrow_{s}\\!x\in\sigma_{s}(L)$ and if
$U\in\sigma_{s}(L)$, then $U=\bigcup\\{\Uparrow_{s}\\!u:u\in U\\}$;

  3. (3)

$\textrm{SOFilt}(L)$ is a basis of $\sigma_{s}(L)$ and $\sigma_{s}(L)$ is a
continuous lattice;

  4. (4)

$\sigma_{s}(L)$ has enough co-primes and is a continuous lattice;

  5. (5)

$\sigma_{s}(L)$ is completely distributive;

  6. (6)

Both $\sigma_{s}(L)$ and $\sigma_{s}(L)^{\textrm{op}}$ are continuous.

If $L$ is a complete semilattice, then these conditions are equivalent to

  1. (7)

For each point $x\in L$, $\Uparrow_{s}\\!x\in\sigma_{s}(L)$ and
$x=\textrm{sup}\\{\textrm{inf}\ \\!U:x\in U\in\sigma_{s}(L)\\}$.

######  Proof 4.15.

$(1)\Leftrightarrow(5)$: By Proposition 2.4 and Definition 3.1.
$(1)\Rightarrow(2)$: By Proposition 4.4 $(1)$.
$(2)\Rightarrow(1)$: Let $x\in L$. If $u\ll_{s}x$, $v\ll_{s}x$, then there
exists $w\in\ \\!\Uparrow_{s}\\!u\ \\!\cap\Uparrow_{s}\\!v$ such that $x\in\
\\!\Uparrow_{s}\\!w$. Thus $\Downarrow_{s}\\!x$ is directed. Set
$y=\bigvee\Downarrow_{s}\\!x\leq x$, if $y<x$, then $L\setminus\downarrow\\!y$
is a strong Scott topology open neighborhood of $x$. Hence there exists $z\in
L\setminus\downarrow\\!y$ such that $x\in\ \\!\Uparrow_{s}\\!z$, then
$z\ll_{s}x$, and thus $z\leq\bigvee\Downarrow_{s}\\!x=y$, which is a
contradiction.
$(3)\Leftrightarrow(4)$: By Proposition 4.8 $(1)$.
$(4)\Leftrightarrow(5)\Leftrightarrow(6)$: By Theorem 2.6.
$(3)\Rightarrow(7)$: Suppose that $L$ is a complete semilattice, then for each
$U\in\sigma_{s}(L)$ and each $x\in U$, $\textrm{inf}\ \\!U$ exists. Let
$y=\textrm{sup}\\{\textrm{inf}\ \\!U:x\in U\in\sigma_{s}(L)\\}$. Then $y\leq
x$. Assume that $x\nleq y$, then $x\in L\setminus\downarrow
y\in\sigma_{s}(L)$. Since $\sigma_{s}(L)$ is a continuous lattice, there
exists $V\in\sigma_{s}(L)$ such that $x\in V\ll L\setminus\downarrow y$. And
by the conditions, there exists $U\in\textrm{SOFilt}(L)$ such that $x\in
U\subseteq V\ll L\setminus\downarrow y$. Note that $\textrm{inf}\ \\!U\leq y$,
hence $L\setminus\downarrow y\subseteq L\setminus\downarrow\textrm{inf}\
\\!U=L\setminus\bigcap\\{\downarrow u:u\in U\\}=\bigcup\\{L\setminus\downarrow
u:u\in U\\}$. Since $U$ is a filter, $\\{L\setminus\downarrow u:u\in U\\}$
form a directed family of strong Scott topology open sets. Since $U\ll
L\setminus\downarrow y$, there is a $u\in U$ with $U\subseteq
L\setminus\downarrow u$, which is a contradiction. Thus $x\leq y$. Since
$(2)\Leftrightarrow(3)$, we have $\Uparrow_{s}\\!x\in\sigma_{s}(L)$ for all
$x\in L$.
$(7)\Rightarrow(1)$: We only need to show that for each $U\in\sigma_{s}(L)$
and each $x\in U$, $\textrm{inf}\ \\!U\ll_{s}x$. For any directed set
$D\subseteq L$ and $z\in L$, if $\uparrow\\!\bigvee\\!D\
\\!\cap\uparrow\\!z\subseteq\ \\!\uparrow\\!x$, since $x\in
U\in\sigma_{s}(L)$, there exists $V\in\sigma^{s}(L)$ such that $x\in
V\subseteq U$, and hence $\uparrow\\!\bigvee\\!D\
\\!\cap\uparrow\\!z\subseteq\ \\!\uparrow\\!x\subseteq V$. Thus there is a
$d\in D$ with $\uparrow\\!d\ \\!\cap\uparrow\\!z\subseteq V$, then
$\uparrow\\!d\ \\!\cap\uparrow\\!z\subseteq V\subseteq U\subseteq\
\\!\uparrow\textrm{inf}\ \\!U$. Therefore $\textrm{inf}~{}U\ll_{s}x$.

######  Corollary 4.16.

If $L$ is a complete semilattice, then $L$ is hypercontinuous iff
$x=\sup\\{\inf U:x\in U\in\upsilon(L)\\}$ for all $x\in L$.

######  Proof 4.17.

By Theorem 3.14 and Theorem 4.14.

Now we introduce the notation of strong Lawson topology and discuss the
properties of strongly continuous domains endowed with the strong Lawson
topology.

######  Definition 4.18.

Let $L$ be a dcpo. The common refinement $\sigma_{s}(L)\vee\omega(L)$ of the
strong Scott topology and the lower topology is called the _strong Lawson
topology_ and is denoted by $\lambda_{s}(L)$.

######  Lemma 4.19.

([2]) Let $L$ be a complete semilattice. Then $\lambda(L)$ is a compact
$T_{1}$ topology.

######  Theorem 4.20.

Let $L$ be a complete semilattice. Then $\lambda_{s}(L)$ is a compact $T_{1}$
topology.

######  Proof 4.21.

By Lemma 4.19, $(L,\lambda_{s}(L))$ is compact. For each $x\in L$, $\\{x\\}=\
\\!\downarrow\\!x\ \\!\cap\uparrow\\!x$. Since $\downarrow\\!x$ is a strong
Scott topology closed set and $\uparrow\\!x$ is a lower topology closed set,
$\\{x\\}$ is a strong Lawson topology closed set. Thus $(L,\lambda_{s}(L))$ is
a $T_{1}$ space.

######  Theorem 4.22.

Let $L$ be a strongly continuous domain. Then $(L,\lambda_{s}(L))$ is a
$T_{2}$ space.

######  Proof 4.23.

Let $x,y\in L$, and $x\neq y$, assume that $x\nleq y$, then there exists
$u\ll_{s}x$, such that $u\nleq y$. Thus $x\in\
\\!\Uparrow_{s}\\!u\in\sigma_{s}(L)$, $y\in
L\setminus\uparrow\\!u\in\omega(L)$, and
$\Uparrow_{s}\\!u\cap(L\setminus\uparrow\\!u)=\emptyset$. Therefore
$(L,\lambda_{s}(L))$ is a $T_{2}$ space.

######  Corollary 4.24.

Let $L$ is a strongly continuous complete semilattice. Then $\lambda_{s}(L)$
is compact and Hausdorff.

######  Proof 4.25.

By Theorem 4.20 and Theorem 4.22.

Recall that the definition of Scott-continuous functions. For a function $f$
from a dcpo $P$ into a dcpo $Q$, if $f$ is continuous with respect to the
Scott topologies, that is, $f^{-1}(U)\in\sigma(P)$ for all $U\in\sigma(Q)$,
then $f$ is called a _Scott-continuous function_.
A function $f$ is called _strongly Scott-continuous_ if it is continuous with
respect to the strong Scott topologies. Next we discuss the properties of
strongly Scott-continuous functions.

######  Proposition 4.26.

Let $P$, $Q$ be dcpos and $f:P\rightarrow Q$. Consider the following
conditions:

  1. (1)

$f:(P,\sigma_{s}(P))\rightarrow(Q,\sigma_{s}(Q))$ is continuous;

  2. (2)

$f^{-1}(U)\in\sigma_{s}(P)$ for all $U\in\sigma^{s}(Q)$;

  3. (3)

$f^{-1}(U)\in\sigma^{s}(P)$ for all $U\in\sigma^{s}(Q)$;

  4. (4)

For each directed set $D\subseteq P$ and $x\in P$, $\uparrow\\!f(\bigcap_{d\in
D}\uparrow\\!d\ \\!\cap\uparrow\\!x)=\bigcap_{d\in D}\uparrow\\!f(d)\
\\!\cap\uparrow\\!f(x)$.

Then $(4)\Rightarrow(3)\Rightarrow(2)\Leftrightarrow(1)$; if both $P$ and $Q$
are sup semilattices and $f$ preserves finite sups, then $(1)\Rightarrow(4)$,
and the four conditions are equivalent.

######  Proof 4.27.

$(1)\Leftrightarrow(2)$: Trivial.
$(4)\Rightarrow(3)$: First we show that $f$ is order preserving. Suppose that
$x\leq y$. Let $D=\\{y\\}$. Then $\uparrow\\!f(\uparrow\\!y\
\\!\cap\uparrow\\!x)=\ \\!\uparrow\\!f(y)\ \\!\cap\uparrow\\!f(x)$. Hence
$\uparrow\\!f(\uparrow\\!y\ \\!\cap\uparrow\\!x)=\
\\!\uparrow\\!f(\uparrow\\!y)$, then $f(y)\in\ \\!\uparrow\\!f(y)\
\\!\cap\uparrow\\!f(x)\subseteq\ \\!\uparrow\\!f(x)$. Thus $f(x)\leq f(y)$.
For each $U\in\sigma^{s}(Q)$, since $f$ is order preserving, $f^{-1}(U)=\
\\!\uparrow\\!f^{-1}(U)$. Suppose directed set $D\subseteq P$ and $x\in P$
such that $\bigcap_{d\in D}\uparrow\\!d\ \\!\cap\uparrow\\!x\subseteq
f^{-1}(U)$, then $f(\bigcap_{d\in D}\uparrow\\!d\
\\!\cap\uparrow\\!x)\subseteq U$, hence $\bigcap_{d\in D}\uparrow\\!f(d)\
\\!\cap\uparrow\\!f(x)=\ \\!\uparrow\\!f(\bigcap_{d\in D}\uparrow\\!d\
\\!\cap\uparrow\\!x)\subseteq U$. Since $U\in\sigma^{s}(Q)$, there is a $d\in
D$ with $\uparrow\\!f(d)\ \\!\cap\uparrow\\!f(x)\subseteq U$. Thus
$\uparrow\\!d\ \\!\cap\uparrow\\!x\subseteq f^{-1}(U)$. Therefore
$f^{-1}(U)\in\sigma^{s}(P)$.
$(3)\Rightarrow(1)$: For each $U\in\sigma_{s}(Q)$, there exists $\\{U_{i}:i\in
I\\}\subseteq\sigma^{s}(Q)$ such that $U=\bigcup_{i\in I}U_{i}$ by the
definition of $\sigma_{s}(Q)$. Then $f^{-1}(U)=f^{-1}(\bigcup_{i\in
I}U_{i})=\bigcup_{i\in I}f^{-1}(U_{i})$. By precondition $(2)$,
$f^{-1}(U_{i})\in\sigma^{s}(P)$ for all $i\in I$. Thus
$f^{-1}(U)\in\sigma_{s}(P)$.
$(1)\Rightarrow(4)$: Suppose that both $P$ and $Q$ are sup semilattices and
$f$ preserves finite sups, then
$\sigma_{s}(P)=\sigma(P),\sigma_{s}(Q)=\sigma(Q)$. For each directed set
$D\subseteq P$ and $x\in P$, we have $\bigcap_{d\in D}\\!\uparrow\\!d\
\\!\cap\uparrow\\!x=\bigcap_{d\in D}\uparrow\\!(d\vee x)=\
\\!\uparrow\\!\bigvee_{d\in D}(d\vee x)$, thus $\uparrow\\!f(\bigcap_{d\in
D}\\!\uparrow\\!d\ \\!\cap\uparrow\\!x)=\ \\!\uparrow\\!f(\bigvee_{d\in
D}(d\vee x))=\ \\!\uparrow\\!\bigvee_{d\in D}f(d\vee x)=\
\\!\uparrow\\!\bigvee_{d\in D}f(d)\vee f(x)=\bigcap_{d\in
D}\uparrow\\!(f(d)\vee f(x))=\bigcap_{d\in D}\uparrow\\!f(d)\
\\!\cap\uparrow\\!f(x)$.

######  Proposition 4.28.

Let $P$ be a strongly continuous domain. Both
$f:(P,\sigma_{s}(P))\rightarrow(Q,\sigma_{s}(Q))$ and
$g:(Q,\sigma_{s}(Q))\rightarrow(P,\sigma_{s}(P))$ are continuous functions,
and $f\circ g=\mathbf{1}_{Q}$. Then $Q$ is a strongly continuous domain.

######  Proof 4.29.

Let $U\in\sigma_{s}(Q)$ and $y\in U$. Then $g(y)\in f^{-1}(U)$. Since $f$ is
continuous, $f^{-1}(U)\in\sigma_{s}(P)$. Since $P$ is a strongly continuous
domain, there exists $u\in P$ such that
$g(y)\in\textmd{int}_{\sigma_{s}(P)}\\!\uparrow\\!u\subseteq\
\uparrow\\!u\subseteq f^{-1}(U)$, thus $y\in
g^{-1}(\textmd{int}_{\sigma_{s}(P)}\\!\uparrow\\!u)\subseteq\
\uparrow\\!f(u)\subseteq U$. Since $g$ is continuous,
$g^{-1}(\textmd{int}_{\sigma_{s}(P)}\\!\uparrow\\!u)\in\sigma_{s}(Q)$. Hence
$y\in\textmd{int}_{\sigma_{s}(Q)}\\!\uparrow\\!f(u)\subseteq\
\uparrow\\!f(u)\subseteq U$. Therefore $Q$ is a strongly continuous domain.

## References

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https://doi.org/10.48550/arXiv.1903.00615

  * [8] Xu, X. and D. Zhao, _On topological Rudin’s lemma, well-filtered spaces and sober spaces_ , Topology and its Applications 272 (2020).
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  * [9] Xu, X. and D. Zhao, _Some open problems on well-filtered spaces and sober spaces_ , Topology and its Applications 301 (2021)
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  * [10] Xu, X. and J. Yang, _Topological representations of distributive hypercontinuous lattices_ , Chin. Ann. Math. 30B (2) (2009), 199–206.
https://doi.org/10.1007/s11401-007-0316-7

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  * [12] Zhang, W. and X. Xu, _S 2-quasicontinuous posets_, Theoretical Computer Science 572 (2015), 78–85.
https://doi.org/10.1016/j.tcs.2015.01.037

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https://doi.org/10.1016/j.jlamp.2014.10.003


J1633+4718 | 7.64 $\pm$ 0.02 | 4.97 $\pm$ 0.02 | 2.67 $\pm$ 0.06 | 27.79 $\pm$ 0.44 | 3.51 $\pm$ 0.25 | 3.35 $\pm$ 0.42 | 0.22
J1703+4540 | 6.01 $\pm$ 0.02 | 3.42 $\pm$ 0.02 | 2.59 $\pm$ 0.06 | 125.17 $\pm$ 1.73 | 14.75 $\pm$ 0.98 | 14.04 $\pm$ 1.69 | 0.52
J1713+3523 | 7.48 $\pm$ 0.02 | 5.38 $\pm$ 0.03 | 2.11 $\pm$ 0.06 | 32.11 $\pm$ 0.53 | 2.43 $\pm$ 0.20 | 2.32 $\pm$ 0.31 | 0.62
J2242+2943 | 5.78 $\pm$ 0.01 | 3.21 $\pm$ 0.02 | 2.57 $\pm$ 0.06 | 153.99 $\pm$ 1.84 | 17.80 $\pm$ 1.16 | 16.94 $\pm$ 2.02 | 1.16
J2314+2243 | 6.74 $\pm$ 0.02 | 4.40 $\pm$ 0.03 | 2.33 $\pm$ 0.07 | 64.08 $\pm$ 0.94 | 5.93 $\pm$ 0.43 | 5.65 $\pm$ 0.72 | 0.85

111111 (1) Short name, (2) WISE W3-band magnitude, (3) WISE W4-band magnitude,
(4) WISE W3-W4 magnitude colour, (5) WISE W3-band flux density in mJy (6) 20
cm flux density in mJy estimated from the WISE W4-band flux density, using a
relation based on CDFS data (Boyle et al. 2007), (7) 20 cm flux density in mJy
estimated from the WISE W4-band flux density, using a relation based on ELAIS
data (Boyle et al. 2007), (8) $q22$ parameter = log ($S_{22\mu m}$
/$S_{\textrm{1.4\leavevmode\nobreak\ GHz}}$).

Table 8: Summary of the radio and mid-infrared results.

Short name | Morph. | Spectral | Other radio | W3-W4 | $q22$ | $R_{\textrm{CDFS}}$ | $R_{\textrm{W3}}$ | Verdict
---|---|---|---|---|---|---|---|---
| type | type | properties | colour |  |  |  |
J0347+0105 | E | S |  | nSF | SF | ? | SF | SF
J0629-0545 | I | S | J? | SF | SF | ? | SF | mix
J0632+6340 | E | S | O? | SF | SF | SF | SF | mix
J0706+3901 | I | S |  | SF | nSF | AGN | SF? | ?
J0713+3820 | E | S |  | nSF | SF | ? | SF | SF
J0804+3853 | E | S | O? | SF | SF | SF | SF | mix
J0806+7248 | I | S | CSS | SF | nSF | AGN | AGN? | AGN
J0814+5609 | I | F | J | nSF | nSF | AGN | AGN? | AGN
J0913+3658 | I | S |  | SF | SF | SF | SF | SF
J0925+5217 | E | F |  | nSF | SF | AGN? | SF? | mix
J0926+1244 | I | S | J | nSF | SF | SF | SF | mix
J0937+3615 | E | S |  | SF | SF | SF | SF | SF
J0952-0136 | I | S | J | nSF | SF | ? | SF | AGN/mix
J1034+3938 | I | S |  | nSF | nSF | AGN | SF? | mix
J1038+4227 | E | S |  | nSF | nSF | AGN | ? | AGN
J1047+4725 | E | S | CSS | SF | nSF | AGN | AGN? | AGN
J1048+2222 | I | S | O | nSF | SF | AGN | SF | AGN/mix
J1102+2239 | I | S |  | nSF | nSF | AGN | SF | mix
J1110+3653 | I | F | J | N/A | N/A | N/A | N/A | AGN
J1121+5351 | I | S |  | nSF | SF | SF | SF | SF/mix
J1138+3653 | I | S | CSS | SF | nSF | AGN | AGN? | AGN
J1159+2838 | I | S |  | SF | SF | ? | SF | SF
J1203+4431 | E | S | J | nSF | SF | SF | SF | AGN
J1209+3217 | I | S |  | SF | SF | SF | SF | SF
J1215+5442 | I | S |  | nSF | SF | SF | SF | SF
J1218+2948 | E | S |  | SF | SF | SF | SF | mix
J1227+3214 | I | S |  | SF | SF | AGN | SF | mix
J1242+3317 | E | S | J/O | SF | SF | SF | SF | mix
J1246+0222 | E | F |  | SF | SF | SF | SF | mix
J1302+1624 | E | S | J | SF | nSF | AGN | AGN? | AGN
J1305+5116 | E | F | J | nSF | nSF | AGN | AGN? | AGN
J1317+6010 | E | S |  | SF | SF | ? | SF | mix?
J1337+2423a | I | S |  | nSF | SF | SF | SF | mix
J1348+2622 | I | S | CSS | nSF | nSF | AGN | ? | AGN
J1358+2658 | I | F |  | SF | SF | AGN? | SF | mix
J1402+2159 | I | S |  | SF | SF | SF | SF | SF
J1536+5433 | I | F |  | nSF | SF | SF | SF | ?
J1555+1911 | E | S |  | SF | SF | SF | SF | SF
J1559+3501 | E | S |  | nSF | SF | SF | SF | SF
J1633+4718 | I | S | J | SF | nSF | AGN | ? | AGN
J1703+4540 | I | S | J | SF | nSF | AGN | ? | AGN/mix
J1713+3523 | I | S |  | nSF | nSF | AGN | SF | mix?
J2242+2943 | E | S |  | SF | SF | ? | SF | mix
J2314+2243 | I | S | CSS | nSF | nSF | AGN | SF | AGN

121212 (1) Short name, (2) radio morphology type: I = intermediate, E =
extended, same as in Table 7, (3) radio spectral type: F = flat, S = steep,
(4) other radio properties: CSS = CSS-like, J = jet, O = outflow, (5), W3-W4
colour: SF = supports enhanced star formation, nSF = does not require enhanced
star formation, (6) $q22$ parameter, SF and nSF as before, (7)
$S_{\textrm{5.2\leavevmode\nobreak\ GHz}}$ CDFS vs.
$S_{\textrm{5.2\leavevmode\nobreak\ GHz}}$ JVLA: SF =
$S_{\textrm{5.2\leavevmode\nobreak\ GHz}}$ JVLA could be explained by star
formation, AGN = $S_{\textrm{5.2\leavevmode\nobreak\ GHz}}$ JVLA cannot be
explained by star formation, ? = neither of the options is preferred, (8)
$S_{\textrm{W3}}$ WISE vs. $S_{\textrm{1.4\leavevmode\nobreak\ GHz}}$ JVLA: SF
= star formation dominated, AGN = AGN dominated, ? = combination, or no
preferred option, (8) the origin of radio emission based on all data and
diagnostic tools: SF = star formation, AGN = AGN activity, mix = combination
of star formation and AGN activity, ? = inconclusive.

##  Appendix B $\alpha$, $\Delta\alpha$, and tapered maps

(a) Spectral index map, rms = 14$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 7], beam size 0.23 $\times$ 0.17 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
7(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 25$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 7], beam size 0.69 $\times$ 0.63 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 33$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 6], beam size 1.01 $\times$ 0.94 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 7(c)

.

Figure 7:

(a) Spectral index map, rms = 14$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 8], beam size 3.09 $\times$ 0.93 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
8(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 22$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 7], beam size 4.15 $\times$ 2.73 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 26$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 7], beam size 5.12 $\times$ 3.91 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the normal map.
Radio map properties as in Fig. 8(a)

.

(f) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 8(c)

.

Figure 8:

(a) Spectral index map, rms = 10 $\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 5], beam size 0.22 $\times$ 0.11 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
9(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 14$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 5], beam size 0.42 $\times$ 0.34 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 16$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 5], beam size 0.59 $\times$ 0.48 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 9(c)

.

(f) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 9(c)

.

Figure 9:

(a) Spectral index map, rms = 13$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 5], beam size 1.68 $\times$ 0.85 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
10(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 17$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 5], beam size 2.66 $\times$ 2.27 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 21$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 5], beam size 3.53 $\times$ 3.21 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 10(c)

.

Figure 10:

(a) Spectral index map, rms = 13$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 5], beam size 2.45 $\times$ 1.19 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
11(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 19$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 5], beam size 3.65 $\times$ 3.01 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 23$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 5], beam size 4.82 $\times$ 4.33 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 11(c)

.

Figure 11:

(a) Spectral index map, rms = 10$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 3], beam size 1.55 $\times$ 1.42 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
12(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 18$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 3], beam size 4.73 $\times$ 4.59 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 23$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 3], beam size 6.66 $\times$ 6.46 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 12(c)

.

Figure 12:

(a) Spectral index map, rms = 10$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 8], beam size 1.30 $\times$ 0.72 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
13(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 11$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 8], beam size 2.79 $\times$ 2.41 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 12$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 8], beam size 4.08 $\times$ 3.37 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 13(c)

.

Figure 13:

(a) Spectral index map, rms = 10$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 9], beam size 5.12 $\times$ 2.28 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
14(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 16$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 9], beam size 10.05 $\times$ 7.77 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 19$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 8], beam size 14.74 $\times$ 10.98
kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the normal map.
Radio map properties as in Fig. 14(a)

.

(f) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 14(c)

.

Figure 14:

(a) Spectral index map, rms = 11$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 3], beam size 2.31 $\times$ 0.80 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
15(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 17$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 2], beam size 3.47 $\times$ 2.58 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 26$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 2], beam size 4.41 $\times$ 3.82 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 15(c)

.

Figure 15:

(a) Spectral index map, rms = 12$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 5], beam size 0.38 $\times$ 0.33 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
16(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 16$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 5], beam size 1.00 $\times$ 0.95 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 21$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 5], beam size 1.39 $\times$ 1.35 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 16(c)

.

Figure 16:

(a) Spectral index map, rms = 11$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 6], beam size 0.55 $\times$ 0.24 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
17(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 17$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 5], beam size 0.99 $\times$ 0.72 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 22$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 5], beam size 1.42 $\times$ 1.01 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 17(c)

.

Figure 17:

(a) Spectral index map, rms = 12$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 4], beam size 3.92 $\times$ 1.24 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
18(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 15$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 4], beam size 5.68 $\times$ 4.10 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 19$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 4], beam size 6.98 $\times$ 6.07 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 18(c)

.

Figure 18:

(a) Spectral index map, rms = 10$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 9], beam size 0.37 $\times$ 0.17 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
19(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 12$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 8], beam size 0.69 $\times$ 0.53 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 14$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 8], beam size 0.96 $\times$ 0.76 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 19(c)

.

Figure 19:

(a) Spectral index map, rms = 10$\mu$Jy, contour levels at -3, 3 $\times$ 2n,
$n\in$ [0, 7], beam size 0.46 $\times$ 0.36 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
20(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 12$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 7], beam size 1.17 $\times$ 1.08 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 14$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 7], beam size 1.63 $\times$ 1.53 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 20(c)

.

Figure 20:

(a) Spectral index map, rms = 10$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 6], beam size 1.92 $\times$ 1.60 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
21(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 13$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 5], beam size 4.97 $\times$ 4.65 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 17$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 5], beam size 6.96 $\times$ 6.57 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 21(c)

.

Figure 21:

(a) Spectral index map, rms = 35$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 10], beam size 2.78 $\times$ 1.95 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
22(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 50$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 11], beam size 7.66 $\times$ 7.28 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 80$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 10], beam size 11.56 $\times$ 11.03
kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 22(c)

.

Figure 22:

(a) Spectral index map, rms = 8$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 3], beam size 3.14 $\times$ 2.00 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
23(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 9$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 3], beam size 7.27 $\times$ 6.04 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 10$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 3], beam size 10.12 $\times$ 8.55 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 23(c)

.

Figure 23:

(a) Spectral index map, rms = 13$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 4], beam size 3.76 $\times$ 2.37 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
24(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 36$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 2], beam size 8.84 $\times$ 7.40 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 45$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 2], beam size 12.43 $\times$ 10.58
kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 24(c)

.

Figure 24:

(a) Spectral index map, rms = 10$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 8], beam size 3.42 $\times$ 2.80 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
25(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 11$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 7], beam size 9.70 $\times$ 8.81 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 12$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 7], beam size 13.66 $\times$ 12.50
kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the normal map.
Radio map properties as in Fig. 25(a)

.

(f) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 25(c)

.

Figure 25:

(a) Spectral index map, rms = 11$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 4], beam size 0.97 $\times$ 0.85 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
26(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 12$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 5], beam size 2.63 $\times$ 2.50 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 14$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 4], beam size 3.71 $\times$ 3.50 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 26(c)

.

Figure 26:

(a) Spectral index map, rms = 10$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 7], beam size 2.60 $\times$ 2.20 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
27(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 12$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 7], beam size 7.04 $\times$ 6.49 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 12$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 7], beam size 9.94 $\times$ 9.19 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 27(c)

.

Figure 27:

(a) Spectral index map, rms = 10$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 4], beam size 1.78 $\times$ 1.44 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
28(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 13$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 4], beam size 5.00 $\times$ 4.35 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 14$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 4], beam size 7.09 $\times$ 6.13 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 28(c)

.

Figure 28:

(a) Spectral index map, rms = 12$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 4], beam size 0.02 $\times$ 0.02 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
29(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 29$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 4], beam size 0.06 $\times$ 0.05 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 38$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 4], beam size 0.08 $\times$ 0.08 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 29(c)

.

(f) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 29(c)

.

Figure 29:

(a) Spectral index map, rms = 13$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 3], beam size 0.78 $\times$ 0.76 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
30(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 14$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 3], beam size 2.65 $\times$ 2.58 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 15$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 3], beam size 3.97 $\times$ 3.87 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 30(c)

.

Figure 30:

(a) Spectral index map, rms = 10$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 4], beam size 1.20 $\times$ 1.07 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
31(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 12$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 4], beam size 3.56 $\times$ 3.48 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 13$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 4], beam size 4.99 $\times$ 4.91 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 31(c)

.

Figure 31:

(a) Spectral index map, rms = 9$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 8], beam size 0.15 $\times$ 0.12 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
32(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 10$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 8], beam size 0.39 $\times$ 0.34 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 12$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 8], beam size 0.56 $\times$ 0.48 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 32(c)

.

(f) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 32(c)

.

Figure 32:

(a) Spectral index map, rms = 11$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 6], beam size 1.23 $\times$ 1.01 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
33(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 17$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 5], beam size 3.54 $\times$ 3.06 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 23$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 5], beam size 5.03 $\times$ 4.31 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 33(c)

.

Figure 33:

(a) Spectral index map, rms = 12$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 5], beam size 0.43 $\times$ 0.33 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
34(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 16$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 5], beam size 1.38 $\times$ 1.01 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 21$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 5], beam size 2.04 $\times$ 1.41 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 34(c)

.

Figure 34:

(a) Spectral index map, rms = 10$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 3], beam size 0.53 $\times$ 0.42 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
35(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 11$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 4], beam size 1.35 $\times$ 1.22 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 15$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 3], beam size 1.91 $\times$ 1.72 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 35(c)

.

Figure 35:

(a) Spectral index map, rms = 12$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 6], beam size 0.58 $\times$ 0.50 kpc. Non-common
uvrange.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
36(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 28$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 5], beam size 1.94 $\times$ 1.59 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 35$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 5], beam size 2.66 $\times$ 2.27 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 36(c)

.

Figure 36:

(a) Spectral index map, rms = 12$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 6], beam size 0.60 $\times$ 0.54 kpc. Common uvrange.

0.47 x 0.42

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
37(a).

Figure 37:

(a) Spectral index map, rms = 14$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 9], beam size 12.70 $\times$ 3.21 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
38(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 16$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 9], beam size 17.03 $\times$ 8.66 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 20$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 9], beam size 22.18 $\times$ 12.10
kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the normal radio
map. Radio map properties as in Fig. 38(a)

.

(f) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 38(c)

.

Figure 38:

(a) Spectral index map, rms = 16$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 3], beam size 2.20 $\times$ 0.73 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
39(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 24$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 3], beam size 2.98 $\times$ 2.37 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 29$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 3], beam size 4.09 $\times$ 3.39 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 39(c)

.

Figure 39:

(a) Spectral index map, rms = 13$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 7], beam size 2.37 $\times$ 0.91 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
40(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 14$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 7], beam size 3.59 $\times$ 2.59 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 16$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 7], beam size 4.70 $\times$ 3.69 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 40(c)

.

Figure 40:

(a) Spectral index map, rms = 12$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 3], beam size 10.26 $\times$ 3.29 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
41(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 14$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 3], beam size 14.72 $\times$ 10.18
kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 15$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 3], beam size 18.48 $\times$ 14.72
kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 41(c)

.

Figure 41:

(a) Spectral index map, rms = 11$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 3], beam size 6.24 $\times$ 2.00 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
42(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 13$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 3], beam size 8.90 $\times$ 6.48 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 15$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 3], beam size 11.05 $\times$ 9.38 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 42(c)

.

Figure 42:

(a) Spectral index map, rms = 16$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 2], beam size 2.20 $\times$ 0.54 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
43(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 22$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 2], beam size 2.90 $\times$ 1.62 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 22$\mu$Jy beam-1, contour
levels at -3, 3, 6, beam size 3.42 $\times$ 2.33 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 43(c)

.

Figure 43:

(a) Spectral index map, rms = 10$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 3], beam size 0.53 $\times$ 0.33 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
44(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 18$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 3], beam size 1.17 $\times$ 1.00 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 23$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 2], beam size 1.66 $\times$ 1.40 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 44(c)

.

Figure 44:

(a) Spectral index map, rms = 11$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 2], beam size 0.54 $\times$ 0.30 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
45(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 15$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 2], beam size 1.12 $\times$ 0.88 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 17$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 2], beam size 1.57 $\times$ 1.24 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 45(c)

.

Figure 45:

(a) Spectral index map, rms = 11$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 4], beam size 0.91 $\times$ 0.25 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
46(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 14$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 4], beam size 1.26 $\times$ 0.79 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 16$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 4], beam size 1.59 $\times$ 1.13 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 46(c)

.

Figure 46:

(a) Spectral index map, rms = 10$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 9], beam size 1.60 $\times$ 0.84 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
47(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 12$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 9], beam size 3.30 $\times$ 2.63 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 14$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 9], beam size 4.69 $\times$ 3.74 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 47(c)

.

Figure 47:

(a) Spectral index map, rms = 11$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 9], beam size 0.88 $\times$ 0.48 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
48(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 17$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 9], beam size 1.85 $\times$ 1.47 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 22$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 9], beam size 2.62 $\times$ 2.09 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 48(c)

.

Figure 48:

(a) Spectral index map, rms = 10$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 6], beam size 1.16 $\times$ 0.58 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
49(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 13$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 6], beam size 2.45 $\times$ 1.87 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 15$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 6], beam size 3.43 $\times$ 2.65 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 49(c)

.

Figure 49:

(a) Spectral index map, rms = 11$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 7], beam size 0.27 $\times$ 0.24 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
50(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 15$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 7], beam size 0.73 $\times$ 0.66 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 25$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 7], beam size 1.06 $\times$ 0.92 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 50(c)

.

Figure 50:

(a) Spectral index map, rms = 10$\mu$Jy beam-1, contour levels at -3, 3
$\times$ 2n, $n\in$ [0, 7], beam size 1.70 $\times$ 1.27 kpc.

(b) Spectral index error map, rms, contour levels, and beam size as in Fig.
51(a).

(c) Tapered map with uvtaper = 90k$\lambda$, rms = 14$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 7], beam size 4.27 $\times$ 3.69 kpc.

(d) Tapered map with uvtaper = 60k$\lambda$, rms = 15$\mu$Jy beam-1, contour
levels at -3, 3 $\times$ 2n, $n\in$ [0, 7], beam size 6.17 $\times$ 5.19 kpc.

(e) PanSTARRS $i$ band image of the host galaxy overlaid with the 90k$\lambda$
tapered map. Radio map properties as in Fig. 51(c)

.

Figure 51: