Locally closed sets and submaximal spaces

Throughout this paper we consider topological spaces on which no separation axioms are assumed unless explicity stated. The topology of a space is denoted by ${\mathcal{T}}$ and $(X,{\mathcal{T}})$ will be replaced by $X$ if there is no chance for confusion. For a subset $A$ of $(X,{\mathcal{T}})$, the closure, the interior, the boundary and the set of accumulation points of $A$ are denoted by ${\rm cl}_{\mathcal{T}}(A)$ or ${\rm cl}_{X}(A)$, ${\rm int}_{\mathcal{T}}(A)$ or ${\rm int}_{X}(A)$, ${\rm Fr}_{\mathcal{T}}(A)$ or ${\rm Fr}_{X}(A)$ and $A^{\prime}$, respectively. In places where there is no chance for confusion $\overline{A}$ and $A^{\circ}$ stands for ${\rm cl}_{\mathcal{T}}(A)$ and ${\rm int}_{\mathcal{T}}(A)$, respectively. If $A\subseteq X$ be open, closed or locally closed respect to the topology ${\mathcal{T}}$ on $X$, then we write sometimes to avoid confusion, ${\mathcal{T}}$-open, ${\mathcal{T}}$-closed or ${\mathcal{T}}$-locally closed, respectively. A subset $A$ of a topological space $X$ is called locally closed if for every $x\in A$ there is an open set $U\subseteq X$ such that $x\in U$ and $A\cap U$ is closed in $U$. Since the intersection of two locally closed sets is locally closed, the family of ${\mathcal{T}}$-locally closed sets forms a base for a finer topology ${\mathcal{T}_{l}}$ on $X$. For a Tychonoff (completely regular Hausdorff) space $X$, $\beta X$ is the Stone-Čech compactification and $\upsilon X$ is the Hewitt realcompactification of $X$. It is well known that $X$ is compact, pseudocompact or realcompact if and only if $\beta X=X$, $\beta X=\upsilon X$ or $\upsilon X=X$, respectively. For a Tychonoff space the symbol $C(X)$ (resp. $C^{*}(X)$) denotes the ring of all continuous real valued (resp. all bounded continuous real valued) functions defined on $X$. We say that a subspace $S$ of $X$ is $C^{*}$-embedded in $X$ if every function in $C^{*}(S)$ can be extended to a function in $C^{*}(X)$. For $f\in C(X)$, the zero-set of $f$ is the set $Z(f)=\{x\in X:f(x)=0\}$. The set-theoretic complement of $Z(f)$ is denoted by $coz(f)$. The support of a function $f\in C(X)$ is the $\overline{coz(f)}$. For any $p\in\beta X$, the maximal ideal $M^{p}$ (resp. the ideal $O^{p}$) is the set of all $f\in C(X)$ for which $p\in{\rm cl}_{\beta X}Z(f)$ (resp. $p\in{\rm int}_{\beta X}{\rm cl}_{\beta X}Z(f)$). More generally, for $A\subseteq\beta X$, $M^{A}$ (resp. $O^{A}$) is the intersection of all $M^{p}$ (resp. $O^{p}$) with $p\in A$. It is clear that $O^{A}\subseteq M^{A}$.

A space $X$ is said to be a, a) door space if every set is either open or closed; b) $T_{\frac{1}{2}}$-space if every singleton is either open or closed; c) submaximal space if every dense subset is open; d) principal space, if every intersection of open sets is open, e) $c$-principal space, if every countable intersection of open sets (i.e., $G_{\delta}$-set) is open, f) resolvable space if it has two disjoint dense subsets.

Tychonoff $c$-principal spaces are the same $P$-spaces, see Exercise 4J of [12]. A door space is submaximal and a submaximal space is $T_{\frac{1}{2}}$. The one-point compactification of any discrete space is submaximal. A nonempty resolvable space never submaximal. For example, $\mathbb{R}$ and $\beta\mathbb{Q}$ are resolvable spaces. For more information about the locally closed sets, see [2, 8, 11], about the submaximal spaces, see [2, 8, 7, 1], about the door spaces, see [7, 9] and about the $T_{\frac{1}{2}}$-spaces, see [3]. For details about $\beta X$ and $\upsilon X$, see [12, 18] and about other concepts of general topology, see [10, 19].

Section 2, of this paper is devoted to investigate some separation properties between $(X,{\mathcal{T}})$ and $(X,{\mathcal{T}_{l}})$. Such as we show that $(X,{\mathcal{T}_{l}})$ is discrete if and only if $(X,{\mathcal{T}})$ is a $T_{D}$-space. We show that, in this section, if $\beta X$ is submaximal, then $X$ is compact and therefor, in this case, we conclude that $X=\beta X$. We also prove that if $\upsilon X$ is submaximal and first countable and $X$ is without isolated point, then $X$ is realcompact. We observe that every submaximal Hausdorff space is pseudo-finite. It turns out that if $\upsilon X$ is a submaximal space, then $X$ is a pseudo-finite $\mu$-compact space. In Section 3, we study and investigate the behavior locally indiscrete spaces. We observe that $(X,{\mathcal{T}})$ is a locally indiscrete space if and only if ${\mathcal{T}}={\mathcal{T}_{l}}$ if and only if every dense open subset of $X$ is regular open. In Section 4, we introduce some lc-properties such as lc-regular and lc-completely regular and compare them with the concepts regular and completely regular. We prove that every clopen subset of a lc-compact space is lc-compact.

A space $X$ is called a $T_{D}$-space if every singleton is locally closed. It is well known that the space $X$ is submaximal if and only if every subset of $X$ is locally closed, see Theorem 4.2 of [8] and also every subspace of a submaximal space is submaximal, see Theorem 1.1 of [8].

The proof of the following proposition is straightforward.

Given a topological space $(X,{\mathcal{T}})$, the collection of all locally closed subsets of $X$ forms a base for a topology on $X$ which is denotes by ${\mathcal{T}_{l}}$. It is clear that ${\mathcal{T}}\subseteq{\mathcal{T}_{l}}$ and in locally indiscrete spaces we have ${\mathcal{T}}={\mathcal{T}_{l}}$, see Proposition 3.4. In the sequel, we study some topological properties between $(X,{\mathcal{T}})$ and $(X,{\mathcal{T}_{l}})$.

If $(X,{\mathcal{T}})$ is a $T_{1}$-space then $(X,{\mathcal{T}_{l}})$ is discrete. The converse is not true. For example, let $X=\{a,b\}$ and ${\mathcal{T}}=\{\emptyset,\{a\},X\}$. For the converse see the next proposition.

If ${\mathcal{T}}$ be the principal topology on $X$, then every element $x\in X$ has a minimal open neighborhood $M_{x}=\bigcap\{G\in{\mathcal{T}}:x\in G\}$. For details about principal topology, see [17].

If $\prod_{\alpha\in\Lambda}X_{\alpha}$ is a $T_{D}$-space, then $X_{\alpha}$ is a $T_{D}$-space for each $\alpha\in\Lambda$. The converse is true if $I$ is finite. In the next example we see that an arbitrary product of $T_{D}$-spaces need not be $T_{D}$.

In the sequel, by $|A|$ we mean the cardinality of $A$ for every set $A$.

Every $T_{\frac{1}{2}}$-space is $T_{D}$. The converse is not true. For example, let $X=\mathbb{N}$ and ${\mathcal{T}}=\{E_{n}:n=1,\cdots\}\cup\{\emptyset\}$, where $E_{n}=\{n,n+1,\cdots\}$, for any $n\in\mathbb{N}$. Since $\{n\}=E_{n}\cap(\mathbb{N}-E_{n+1})$, we infer that $\mathbb{N}$ is a $T_{D}$-space. It is clear that the set $\{2\}$ neither open nor closed, so $\mathbb{N}$ is not a $T_{\frac{1}{2}}$-space.

Recall that union of two locally closed sets may not be locally closed. In the next example, we see that even if we add a cluster point to a locally closed set, the resulting set may not be locally closed. In other words, if $A$ is locally closed and $x\in\overline{A}-A$, then $A\cup\{x\}$ need not be locally closed.

If $X$ is a submaximal space and $A\subseteq X$, then ${\rm Fr}(A)$ is a discrete subset of $X$. Also if $X$ is a submaximal space and $A$ is a discrete subset of $X$, then $\overline{A^{\prime}}$ is discrete and if in addition $X$ is a $T_{1}$-space, then $A^{\prime}$ is discrete.

The converse of the above theorem is not true, in general. For example, $X=\beta\mathbb{Q}$ is compact, while $\beta X=\beta\mathbb{Q}$ is not a submaximal space. Also, if $X$ is a submaximal space, then $\beta X$ may not be submaximal. For example, $\mathbb{N}$ is submaximal but $\beta\mathbb{N}$ is not submaximal.

By the above lemma the proof of the next corollary is obvious.

A space $X$ is called a pseudo-finite (resp. pseudo-discrete) space if every compact subspace of $X$ is finite (resp. has finite interior). We say that $X$ is real pseudo-finite, if $\upsilon X$ is pseudo-finite. Every pseudo-finite space is pseudo-discrete, but not conversely. For example, we consider the space $\mathbb{Q}$ of rational numbers.

A pseudo-finite Tychonoff space may be not submaximal. For example, $\mathbb{Q}$ is pseudo-finite but not submaximal. See also the following example.

$C_{K}(X)$ (resp. $C_{\psi}(X)$) is the family of all $f\in C(X)$ with compact (resp. pseudocompact) support. Also $C_{\infty}(X)$ is a subcollection of $C^{*}(X)$ consisting of all functions vanishing at infinity, that is all $f\in C(X)$ for which $\{x\in X:|f(x)|\geq\frac{1}{n}\}$ is compact for every $n\in\mathbb{N}$. For the convenience of readers, some special ideals are listed below.

a) $C_{F}(X)=O^{\beta X\setminus I(X)}$ is the intersection of all essential ideals in $C(X)$. Recall that $C_{F}(X)$ is the socle of $C(X)$ and it is well known that $C_{F}(X)=\{f\in C(X):coz(f)~{}\mbox{is finite}\}$, see Proposition 3.3 of [14].

b) $C_{K}(X)=O^{\beta X\setminus X}$ is the intersection of all free ideals in $C(X)$, see 7E of [12].

c) $C_{\infty}(X)=M^{*\beta X\setminus X}$ is the intersection of all free maximal ideals in $C^{*}(X)$.

d) $C_{\psi}(X)=M^{\beta X\setminus\upsilon X}$ is the intersection of all hyper-real maximal ideals in $C(X)$.

e) $I_{\psi}(X)=M^{\beta X\setminus X}$ is the intersection of all free maximal ideals in $C(X)$.

One can easily see that $C_{F}(X)\subseteq C_{K}(X)\subseteq I_{\psi}(X)$. In Theorem 8.19 of [12], it is shown that if $X$ is realcompact, then $C_{K}(X)=I_{\psi}(X)$. In part (a) of Theorem 4.5 of [4], it is proved that $X$ is a pseudo-discrete if and only if $C_{F}(X)=C_{K}(X)$.

In [13] a space $X$ is called:
a) a $\mu$-compact if $C_{K}(X)=I_{\psi}(X)$.
b) an $\eta$-compact if $C_{\psi}(X)=I_{\psi}(X)$.
c) a $\psi$-compact if $C_{K}(X)=C_{\psi}(X)$.
d) an $\infty$-compact if $C_{K}(X)=C_{\infty}(X)$.

In [15] a space $X$ is called an $i$-compact if $C_{\infty}(X)=I_{\psi}(X)$.

The converse of the above proposition is not true, in general. For example, $\mathbb{Q}$ is a pseudo-finite $\mu$-compact space but $\upsilon\mathbb{Q}=\mathbb{Q}$ is not submaximal.
By part (b) of Theorem 4.5 of [4], the proof of the result is obvious.

Similar to Theorem 2.16, the question arises that if $\upsilon X$ is a submaximal space, is $X$ realcompact?. So far, we have not been able to answer this question, in general. Therefor we express it as a question.

Question: If $\upsilon X$ is a submaximal space, is $X$ realcompact?

The space $\mathbb{R}$ is a realcompact space which is not submaximal. See the following example for an example of a space that is submaximal but not realcompact. This example, also shows that $X$ may be submaximal but $\upsilon X$ may not be submaximal.

It is possible that $X$ is a locally compact and pseudocompact space but $\upsilon X$ is not submaximal. See the next example.

A space $X$ is called locally indiscrete if every open set is closed or equivalently if every closed set is open. Every discrete space is locally indiscrete. For another nontrivial example, let $R$ be a principal ideal ring. Then the space ${\rm Min}(R)$ with Zariski topology is a locally indiscrete space.

To see the definition of the concepts given in the next remark, see [12, 10]. Foe details about $P_{F}$-spaces, see [6].

In this section, by using the locally closed sets, we introduce some separation axioms. For more details about lc-continuous functions, see [11]. We begin with the following definition.

Every lc-regular (resp. lc-completely regular, lc-normal) space is a regular (resp. completely regular, normal) space. The converse is hold in locally indiscrete spaces, but is not true, in general. See the following example.

Every lc-compact space is compact. Every infinite compact $T_{1}$-space is not lc-compact. One can easily see that $(X,{\mathcal{T}})$ is lc-compact if and only if $(X,{\mathcal{T}_{l}})$ is compact.