Continuous Selections and Extremally Disconnected Spaces

Abstract

This paper deals with extremally disconnected spaces and extremally disconnected P-spaces. A space X is said to be extremally disconnected if, for every open subset V of X, the closure of V in X is also an open set. P-spaces are spaces in which the intersection of countably many open sets is an open set. The authors present a new characterization of extremally disconnected spaces, and the extremally disconnected P-spaces, by means of selection theory. If X is either an extremally disconnected space or an extremally disconnected P-space, then the usual theorems of extension of real-valued continuous functions for a dense subset S of X can be deduced from our results. A corollary of our outcomes is that every nondiscrete space X of nonmeasurable cardinality has a dense subset S such that S is not C-embedded in X.

1. Introduction

Throughout the paper, the word space means a Tychonoff space; that is, a completely regular Hausdorff space. The theory of set-valued functions is an active field of research with multiple applications. It has been developed in a variety of contexts, and it has an extensive bibliography. Its applications range from Functional Analysis to the theory of discrete dynamical systems, passing through the fixed point theory. Among others, the interested reader is referred to [1,2,3,4,5,6,7,8]. In this paper, we deal with selection theory, as introduced by Michael in his classic paper [9]: related to one of the most important subjects in topology, the problem of extension of continuous functions. In this framework, the central notion is the concept of a selection for a set-valued function.

Recall that a set-valued function $\phi$ is a function from a space X into the family $2^Y$ of all nonempty subsets of a space Y. A selection for $\phi$ is a continuous function $f:X\to Y$ such that $f\left(x\right)\in \phi \left(x\right)$ for all $x\in X$. As Michael mentions in his paper above, the problem of extension of continuous functions can be rethought of in the following way:

S.1

Let X and Y be two spaces and consider $2^Y$ equipped with a topology $\mathcal{T}$. If $A\subset X$, then, given a set-valued function $\phi :X\to 2^Y$, under what conditions does a selection f for the restriction of $\phi$ to a A have an extension to a selection $\widehat{f}$ for $\phi$?

S.2

Let X and Y be two spaces and consider $2^Y$ equipped with a topology $\mathcal{T}$. If $A\subset X$, then, given a set-valued function $\phi :X\to 2^Y$ and an open subset U containing A, under what conditions does a selection f for the restriction of $\phi$ to A have an extension to a selection $\widehat{f}$ for $\phi :U\to 2^Y$?

In this context, Michael [9] characterized normal spaces, paracompact spaces, collectionwise normal spaces and normal countable paracompact spaces. As a consequence, he obtained the well-known extension theorems of Urysohn, Dugundji, Dowker and Hanner.

In the spirit of Michael’s theorems, we obtain a new characterization of extremally disconnected spaces and extremally disconnected P-spaces by means of selection theory. As a consequence, we obtain the extension theorems for a dense subset of these kinds of spaces (see Corollaries 1 and 3). Moreover, we show that every nondiscrete space X of nonmeasurable cardinality has a dense subset S such that S is not C-embedded in X. It is worth mentioning that Michael used the so-called lower semicontinuous set-valued functions. However, we establish our results by means of continuous set-valued functions, and we provide examples showing that our characterizations fail to be true for lower semicontinuous set-valued functions.

2. Preliminaries

A space X is called extremally disconnected if, for every open subset V of X, the closure in X of V is also open. A compact extremally disconnected space is sometimes called a Stonean space (see [10] for some interesting applications of Stonean spaces). The well-known Semadeni’s theorem states that a compact space X is a Stonean space if and only if X is a retract of the Stone–Čech compactification of a discrete space ([11] Theorem 24.7.1). In the duality between Stone spaces (totally disconnected compact spaces) and Boolean algebras, Stonean spaces correspond to complete Boolean algebras (the interested reader can consult [12] Ch. 5). Examples of extremally disconnected spaces are: discrete spaces, the Stone-Čech compactification of a discrete space, the spectrum of an abelian von Neumann algebra, etc. Recall that a space X is said to be a P-space if every countable intersection of open sets of X is open in X. This kind of spaces were introduced by Gillman and Henriksen in [13]. Examples of P-spaces are Alexandroff-discrete (finitely generated) spaces [14].

If $(Y,d)$ is a metric space and $A\subset Y$, given $\epsilon >0$, let $B_{\epsilon }\left(A\right)$ denote the set $\{y\in Y:d(y,A)<\epsilon \}$, where $d(y,A)=\inf \left\{d\right(y,z):z\in A\}$. As usual, the ball of center a point $x\in Y$ and radius $\epsilon$ will be denoted by $B_{\epsilon }\left(x\right)$; that is, $B_{\epsilon }\left(x\right)=\{y\in Y:d(x,y)<\epsilon \}$. A set-valued function $\phi :X\to \mathcal{F}\subset 2^Y$ is said to be continuous if it is continuous when $\mathcal{F}$ is endowed with the topology induced by the Hausdorff metric; that is, if for every $\epsilon >0$ and every $x_0\in X$, there is a neighborhood U of $x_0$ such that

$$\phi \left(x_0\right)\subset B_{\epsilon }\left(\phi \left(x\right)\right)\mathrm{and}\phi \left(x\right)\subset B_{\epsilon }\left(\phi \left(x_0\right)\right)$$

for all $x\in U$.

A set-valued function $\phi :X\to \mathcal{F}\subset 2^Y$ is called lower semicontinuous if for every open subset V of Y, the set $\{x\in X:\phi (x)\cap V\ne \varnothing )\}$ is open in X. In other words, $\phi$ is continuous when $\mathcal{F}$ is endowed with the topology that has as a subbase the sets of the form $\{B_V:V\mathrm{is}\mathrm{an}\mathrm{open}\mathrm{set}\mathrm{in}Y\}$, where $B_V=\{A\in \mathcal{F}:A\cap V\ne \varnothing \}$.

Our terminology and notation are standard. For example, $\mathbb{N}$ stands for the natural numbers and $\mathbb{R}$ for the reals. The unit interval is denoted by $\mathbb{I}$. When considered as topological spaces, all of them are equipped with their usual topology. $\beta \left(X\right)$ (respectively, $\upsilon \left(X\right)$) means the Stone-Čech compactification (respectively, the Hewitt realcompactification) of X. The restriction of a function f to a subset A is denoted by ${f|}_A$, and the closure (in X) of a subset A of X by ${\mathrm{cl}}_{X}A$. A subset Z of a space X is called a zero set in X if there exists a real-valued continuous function f on X such that $Z\equiv Z\left(f\right)=\{x\in X:f(x)=0\}$. Notice that every zero set is a $G_{\delta }$-set, i.e., a countable intersection of open sets. A subset A of a space X is named C-embedded (respectively, $C^{\star }$-embedded) in X if every real-valued (respectively, every bounded real-valued) continuous function on A has a continuous extension to the whole X. Given a space X, a net $\{x_i:i\in I\}$ is said to be a universal net or an ultranet if for every $A\subset X$, $\{x_i:i\in I\}$ is eventually in A or in $X\setminus A$. There are well-known facts that an ultranet clusters at a point $x\in X$ if and only if it converges to x, and that every net contains a subnet that is an ultranet. For notions not defined here, the reader can consult [15].

3. The Results

Our first result characterizes extremally disconnected spaces by using selection theory. The following lemma will be helpful in the sequel.

Lemma 1.

Let $f:S\to Y$ be a continuous function from a dense subset S of an extremally disconnected space X into a metric space $(Y,d)$. Let $\{x_i:i\in I\}$ and $\{x_h:h\in H\}$ be two nets in S which converge to a point $x\in X$. If $\{f\left(x_i\right):i\in I\}$ converges to a point $y\in Y$, then so does $\{f\left(x_h\right):h\in H\}$.

Proof. 

Consider two nets $\{x_i:i\in I\}$ and $\{x_h:h\in H\}$ in S converging to $x\in X$ such that the limit of $\{f\left(x_i\right):i\in I\}$ is the point $y\in Y$. Suppose now that $\{f\left(x_h\right):h\in H\}$ does not converge to y. Then we can assume, with no loss of generality, that there is $\epsilon >0$ such that $d(f\left(x_h\right),y)\ge \epsilon$ for all $h\in H$, and we can select $i_0\in I$ with $d(f\left(x_i\right),y)<\epsilon /2$ for all $i\ge i_0$.

Consider now, the disjoint open sets A and B in S defined, respectively, as $f^{-1}\left(B_{\frac{\epsilon }{2}}\left(y\right)\right)$ and $f^{-1}(Y\setminus {\mathrm{cl}}_Y{B}_{\frac{\epsilon }{2}}\left(y\right))$. If V and W are open sets in X with $A=V\cap S$ and $B=W\cap S$, then $x\in {\mathrm{cl}}_{X}A\cap {\mathrm{cl}}_{X}B\subset {\mathrm{cl}}_{X}V\cap {\mathrm{cl}}_{X}W.$ We will see that this leads us to a contradiction. Since X is extremally disconnected, ${\mathrm{cl}}_{X}V$ is an open set and, since $x\in {\mathrm{cl}}_{X}W$, there is $w\in {\mathrm{cl}}_{X}V\cap W$. This implies that $V\cap W\ne \varnothing$. By density, $A\cap B\ne \varnothing$ and we have the required contradiction. □

The theorem to be proved is the following. If $(Y,d)$ is a metric space, $\mathcal{K}\left(Y\right)$ denotes the family of all nonempty compact subsets of $(Y,d)$.

Theorem 1.

For a space X, the following conditions are equivalent:

(i) 

X is extremally disconnected.

(ii) 

For any metric space $(Y,d)$, if S is a dense subset of X and $\phi :X\to \mathcal{K}\left(Y\right)$ is a continuous set-valued function, then every selection f for ${\phi |}_S$ has an extension to a selection $\widehat{f}$ for $\phi :X\to \mathcal{K}\left(Y\right)$.

(iii) 

If S is a dense subset of X and $\phi :X\to \mathcal{K}\left(\mathbb{R}\right)$ is a continuous set-valued function, then every selection f for ${\phi |}_S$ has an extension to a selection $\widehat{f}$ for $\phi :X\to \mathcal{K}\left(\mathbb{R}\right)$.

Proof. 

(i)⟹(ii) Let f be a selection for ${\phi |}_S$. Consider, now, $x\in X\setminus S$. By density, there is a net $\{x_i:i\in I\}\subset S$ converging to x. There is no loss of generality in assuming that $\{x_i:i\in I\}$ is an ultranet.

Next, for each $i_0\in I$, we consider the set $A_{i_0}=\{f\left(t_i\right):i\ge i_0\}$. We will prove that ${\mathrm{cl}}_Y{A}_{i_0}\cap \phi \left(x\right)\ne \varnothing$ for all $i_0\in I$. Suppose that, contrary to our claim, there is $i_0\in I$ such that ${\mathrm{cl}}_Y{A}_{i_0}\cap \phi \left(x\right)=\varnothing$. Since Y is a metric space, it is normal. Thus, we can choose pairwise disjoint open subsets U and V of Y such that ${\mathrm{cl}}_Y{A}_{i_0}\subset U$ and $\phi \left(x\right)\subset V.$ The compactness of $\phi \left(x\right)$ tells us that there is $r>0$ such that $B_r\left(z\right)\subset V$ for every $z\in \phi \left(x\right)$ ([16] Theorem 5.26).

Given $\epsilon >0$, we can now apply the continuity of $\phi$ in order to choose $i_1\in I$ such that $\phi \left(x_i\right)\in B_r\left(\phi \left(x\right)\right)$ for all $i\ge i_1$. Then, since f is a selection for ${\phi |}_S$, the previous fact implies that, for all $i\ge i_1$, there exists $z_i\in Y$ such that $d(f\left(x_i\right),z_i)<\epsilon$. Take now $i_2$ with $i_2\ge i_1$ and $i_2\ge i_0$. Then $f\left(x_{i_2}\right)\in {\mathrm{cl}}_Y{A}_{i_1}$ and there is $z_{i_2}\in \phi \left(x\right)$ such that $d(f(x_{i_2},z_{i_2})<\epsilon$. This fact contradicts that the open sets U and V are disjointed. Thus, ${\mathrm{cl}}_Y{A}_i\cap \phi \left(x\right)\ne \varnothing$ for all $i\in I$.

Next, since I is a directed set, the family $\left\{{\mathrm{cl}}_Y{A}_i\cap \phi \left(x\right):i\in I\right\}$ has the finite intersection property. Being $\phi \left(x\right)$ compact, there is $y_x\in {\bigcap }_{i\in I}{\mathrm{cl}}_Y{A}_I\cap \phi \left(x\right)$. In other words, $y_x$ is a cluster point of the ultranet $\{f\left(x_i\right):i\in I\}$ and, consequently, $\{f\left(x_i\right):i\in I\}$ converges to $y_x$. By the previous lemma, the image of every net in S converging to x (in X) is a net that has the limit $y_x$. According to ([15] Problem 6H), the function

$$\widehat{f}\left(x\right)=\left\{\begin{array}{cc}f\left(x\right),\hfill & x\in S,\hfill \\ y_x,\hfill & x\in X\setminus S,\hfill \end{array}\right.$$

is a continuous extension of f to the whole X. It is clear that $\widehat{f}$ is a selection for $\phi$.

(ii)⟹(iii) is obvious. We finish the proof by showing (iii)⟹(i). Let U be an open set in X. We have to prove that ${\mathrm{cl}}_{X}U$ is open. To see this, suppose, with no loss of generality, that $U\ne {\mathrm{cl}}_{X}U$. Consider now the (proper) dense (in X) set $S=U\cup (X\setminus {\mathrm{cl}}_{X}U)$ and define a continuous set-valued function $\phi$ from X into $\mathcal{K}\left(R\right)$ as $\phi \left(x\right)=\mathbb{I}$ for every $x\in X$. Let $f:S\to \mathbb{R}$ be the function defined as $f\left(U\right)=0$ and $f(X\setminus {\mathrm{cl}}_{X}U)=1$. It is obvious that f is a selection for ${\phi |}_S$. According to (iii), there exists a continuous extension $\widehat{f}$ of f to the whole X, which is a selection for $\phi$. By density, $\widehat{f}\left({\mathrm{cl}}_{X}U\right)=0$. Since $\widehat{f}(X\setminus {\mathrm{cl}}_{X}U)=1$, there are pairwise disjoint open sets V and W in $\mathbb{R}$ such that

$$(X\setminus {\mathrm{cl}}_{X}U)\subset {\widehat{f}}^{-1}\left(V\right)\mathrm{and}{\mathrm{cl}}_{X}U\subset {\widehat{f}}^{-1}\left(W\right).$$

Thus, ${\mathrm{cl}}_{X}U$ is an open subset of X. The proof is complete. □

The following corollary is straightforward.

Corollary 1 

(Compare ([15] 6M. 2)). A space X is extremally disconnected if and only if every continuous function from a dense subset S of X into a compact metric space $(K,d)$ has a continuous extension to the whole X. In particular, X is extremally disconnected if and only if every dense subset of X is $C^{\star }$-embedded in X.

The following corollary provides a characterization of Stonean spaces.

Corollary 2.

For a compact space X, the following conditions are equivalent:

(i) 

X is a Stonean space.

(ii) 

For any metric space $(Y,d)$, if S is a dense subset of X and $\phi :X\to \mathcal{K}\left(Y\right)$ is a continuous set-valued function, then every selection f for ${\phi |}_S$ has an extension to a selection $\widehat{f}$ for $\phi :X\to \mathcal{K}\left(Y\right)$.

(iii) 

If S is a dense subset of X and $\phi :X\to \mathcal{K}\left(\mathbb{R}\right)$ is a continuous set-valued function, then every selection f for ${\phi |}_S$ has an extension to a selection $\widehat{f}$ for $\phi :X\to \mathcal{K}\left(\mathbb{R}\right)$.

By way of illustration, here is an example showing that Theorem 1 fails to be true for lower semicontinuous set-valued functions.

Example 1.

Consider a point $p\in \beta \left(\mathbb{N}\right)\setminus \mathbb{N}$. Let Γ be the subspace of $\beta \left(\mathbb{N}\right)$ defined as $\mathrm{\Gamma }=\left\{p\right\}\cup \mathbb{N}$. It is a well-known fact (and easy to prove) that Γ is extremally disconnected. It is a straightforward matter to verify that the set-valued function $\phi :\mathrm{\Gamma }\to \mathcal{K}\left(\mathbb{R}\right)$ defined as

$$\phi \left(x\right)=\left\{\begin{array}{cc}\left\{1\right\},& x\in \mathrm{\Gamma }\setminus \mathbb{N},\hfill \\ \mathbb{I},& x\in \mathbb{N},\hfill \end{array}\right.$$

is lower semicontinuous. Notice that the real-valued function from $f:\mathbb{N}\to \mathbb{R}$ defined as $f\left(n\right)=0$ for all $n\in \mathbb{N}$ is a selection for ${\phi |}_{\mathbb{N}}$ which does not have a continuous extension to Γ .

We now turn our attention to extremally disconnected P-spaces. We shall freely use without explicit mention the elementary fact that a space X is a P-space if and only if every zero set in X is open ([15] Theorem 14.29). We need the following lemma. $\mathcal{C}\left(Y\right)$ stands for the family of all nonempty closed subsets of a metric space $(Y,d)$ equipped with the topology induced by the Hausdorff metric.

Lemma 2.

Let $(Y,d)$ be a metric space and let S be a dense subset of an extremally disconnected P-space X. Consider a set-valued function $\phi :X\to \mathcal{C}\left(Y\right)$. If a net $\{x_i:i\in I\}\subset S$ converges to a point $x\in X\setminus S$, then, for every countable sequence ${\left\{i_n\right\}}_{n\ge 1}\subset I$, we have

$$\left(\bigcap _{n\ge 1}{\mathrm{cl}}_Y{A}_{i_n}\right)\cap \phi \left(x\right)\ne \varnothing ,$$

where $A_{i_n}=\{f\left(x_i\right):i\ge i_n\}$ for all $n=1,2,\dots$.

Proof. 

Let ${\mathrm{\Sigma }}_{n\ge 1}{c}_n$ be a convergent series of positive numbers. Consider the bounded real-valued continuous functions $g_n$ ($n\ge 1$) and h on S defined as

$$g_n\left(s\right)=\min (d(f\left(s\right),{\mathrm{cl}}_Y{A}_{i_n}),a_n)\mathrm{for}\mathrm{all}n\in \mathbb{N},\mathrm{and}h\left(s\right)=\min (d(f\left(s\right),\phi \left(x\right)),1)$$

for all $s\in S$.

By Corollary 1 there exist continuous extensions ${\widehat{g}}_n$ of $g_n$ ($n\ge 1$) and $\widehat{h}$ of h, respectively, to the whole X. Notice that, by definition of the functions $g_n$ ($n\ge 1$) and density of S, we have ${\widehat{g}}_n\left(X\right)\subset [0,a_n]$ for all $n\ge 1$. Therefore, the Weierstrass criterion tells us that the function $\widehat{g}\equiv {\mathrm{\Sigma }}_{n\ge 1}{\widehat{g}}_n$ is a continuous function from X into the reals.

Consider, now, the zero sets $Z\left(\widehat{g}\right)$ and $Z\left(\widehat{h}\right)$. We prove that $x\in Z\left(\widehat{g}\right)\cap Z\left(\widehat{h}\right)$. To see this, we fix $i_n$. Then, for every $i\ge i_n$, $d(f\left(x_i\right),{\mathrm{cl}}_Y{A}_{i_n})=0$, i.e., $g_n\left(x_i\right)={\widehat{g}}_n\left(x_i\right)$=0 for all $i\ge i_n$. Taking into account that the net $\{x_i:i\in I\}$ converges to x, we obtain ${\widehat{g}}_n\left(x\right)=0$. Since $i_n$ is arbitrary, we have just shown that $x\in Z\left(\widehat{g}\right)$.

We prove next that $x\in Z\left(\widehat{h}\right)$. First, notice that, since f is a selection for ${\phi |}_S$, we have that $f\left(x_i\right)\in \phi \left(x_i\right)$ for every $i\in I$. Moreover, $\phi$ is a continuous set-valued function so that the net $\{\phi \left(x_i\right):i\in I\}$ converges to $\phi \left(x\right)$ in the topology induced by the Hausdorff metric. Therefore, the net $\{d(f\left(x_i\right),\phi \left(x\right)):i\in I\}$ converges to zero. By the definition of $\widehat{h}$, we have $\widehat{h}\left(x\right)=0$; that is, $x\in Z\left(\widehat{h}\right)$. Hence, $Z\left(\widehat{g}\right)\cap Z\left(\widehat{h}\right)$ is a nonempty zero set. Since X is a P-space, the zero set $Z\left(\widehat{g}\right)\cap Z\left(\widehat{h}\right)$ meets S.

Select now $s\in Z\left(\widehat{g}\right)\cap Z\left(\widehat{h}\right)\cap S$. We show that $f\left(s\right)\in \left({\bigcap }_{n\ge 1}{\mathrm{cl}}_Y{A}_{i_n}\right)\cap \phi \left(x\right)$. To see this, notice that, since $s\in Z\left(\widehat{g}\right)$, we have $g_n=0$ for all $n\ge 1$, i.e., $d(f\left(s\right),{\mathrm{cl}}_Y{A}_{i_n})=0$ for all $i_n$. Thus, $f\left(s\right)\in {\mathrm{cl}}_Y{A}_{i_n}$ for all $n\ge 1$. On the other hand, the fact that $s\in Z\left(\widehat{h}\right)$ tells us that $d\left(f\right(s),\phi (x\left)\right)=0$, i.e., $f\left(s\right)\in \phi \left(x\right)$. This completes the proof. □

We shall now prove the promised theorem of the characterization of extremally disconnected P-spaces. The symbol $\mathcal{S}\left(Y\right)$ stands for the family of all nonempty separable closed subsets of a metric space $(Y,d)$.

Theorem 2.

For a space X, the following conditions are equivalent:

(i) 

X is an extremally disconnected P-space.

(ii) 

For any metric space $(Y,d)$, if S is a dense subset of X and $\phi :X\to \mathcal{S}\left(Y\right)$ is a continuous set-valued function, then every selection f for ${\phi |}_S$ has a continuous extension to a selection $\widehat{f}$ for $\phi :X\to \mathcal{S}\left(Y\right)$.

(iii) 

For any separable metric space $(Y,d)$, if S is a dense subset of X and $\phi :X\to \mathcal{C}\left(Y\right)$ is a continuous set-valued function, then every selection f for ${\phi |}_S$ has a continuous extension to a selection $\widehat{f}$ for $\phi :X\to \mathcal{C}\left(Y\right)$.

(iv) 

If S is a dense subset of X and $\phi :X\to \mathcal{C}\left(\mathbb{R}\right)$ is a continuous set-valued function, then every selection f for ${\phi |}_S$ has a continuous extension to a selection $\widehat{f}$ for $\phi :X\to \mathcal{C}\left(\mathbb{R}\right)$.

Proof. 

By means of Lemmas 1 and 2, (i)⟹(ii) follows from an argument similar to the one used in the proof of Theorem 1. Moreover, if Y is a separable metric space, then $\mathcal{S}\left(Y\right)=\mathcal{C}\left(Y\right)$, so (iii) is a particular case of (ii). Since (iii)⟹(iv) is obvious, we only have to prove (iv)⟹(i). To see this, notice that $\mathcal{K}\left(\mathbb{R}\right)\subset \mathcal{C}\left(\mathbb{R}\right)$ so that Theorem 1 tells us that X is extremally disconnected. We now show that X is a P-space by showing that every zero set in X is open. Let Z be a zero set in X and denote by ${\mathrm{cl}}_X\mathring{Z}$ the closure of the interior of Z. We suppose that ${\mathrm{cl}}_X\mathring{Z}$ is a proper subset of Z and obtain a contradiction. Notice that, since X is an extremally disconnected space, ${\mathrm{cl}}_X\mathring{Z}$ is clopen.

Claim (1). $Z\setminus {\mathrm{cl}}_X\mathring{Z}$ is a zero set in X.

Since ${\mathrm{cl}}_X\mathring{Z}$ is clopen, the function $f:X\to \mathbb{R}$ defined as

$$f\left(x\right)=\left\{\begin{array}{cc}1,& x\in {\mathrm{cl}}_X\mathring{Z},\hfill \\ 0,& x\in X\setminus {\mathrm{cl}}_X\mathring{Z},\hfill \end{array}\right.$$

is continuous. Then, if $Z=Z\left(g\right)$, we have $Z\setminus {\mathrm{cl}}_X\mathring{Z}=Z\left(h\right)$ with $h=f+g$.

Claim (2). $X\setminus (Z\setminus {\mathrm{cl}}_X\mathring{Z})$ is dense in X. Suppose, contrary to what we claim, that $X\setminus (Z\setminus {\mathrm{cl}}_X\mathring{Z})$ is not a dense subset of X. Then there exists $x\in X$ and an open set V in X with $x\in V$ such that $V\subset Z\setminus {\mathrm{cl}}_X\mathring{Z}$. Therefore, $x\in \mathring{Z}\subset {\mathrm{cl}}_X\mathring{Z}$, a contradiction. Hence, $X\setminus (Z\setminus {\mathrm{cl}}_X\mathring{Z})$ is dense in X. This proves the claim.

Next, we define a set-valued continuous function $\phi$ from X into $\mathcal{C}\left(\mathbb{R}\right)$ as $\phi \left(x\right)=\mathbb{R}$ for all $x\in X$. Consider now the real-valued continuous function p on $X\setminus (Z\setminus {\mathrm{cl}}_X\mathring{Z})$ defined as $p\left(x\right)={\displaystyle \frac{1}{h\left(x\right)}}$ for all $x\in X\setminus (Z\setminus {\mathrm{cl}}_X\mathring{Z})$, where h is the function defined in Claim 1. By Claim (1), p is well-defined. Since by Claim (2) $X\setminus (Z\setminus {\mathrm{cl}}_X\mathring{Z})$ is dense in X, and p is a selection for ${\phi |}_{X\setminus (Z\setminus {\mathrm{cl}}_X\mathring{Z})}$, clause (iv) tells us that there is a selection, say $\widehat{p}$, for $\phi$, which is an extension of p.

Consider now the continuous product function $m=\widehat{p}h$. The restriction of m to $X\setminus (Z\setminus {\mathrm{cl}}_X\mathring{Z})$ is the constant function equal to one and its restriction to $Z\setminus {\mathrm{cl}}_X\mathring{Z}$ is the constant function equal to zero. We have just proven that $X\setminus (Z\setminus {\mathrm{cl}}_X\mathring{Z})$ and $Z\setminus {\mathrm{cl}}_X\mathring{Z}$ are completely separated. This fact leads us to the desired contradiction because, by Claim (2), $X\setminus (Z\setminus {\mathrm{cl}}_X\mathring{Z})$ is dense in X. Thus, $Z={\mathrm{cl}}_X\mathring{Z}$. Since X is extremally disconnected, ${\mathrm{cl}}_X\mathring{Z}$ is an open set in X, and the proof is complete. □

As a straightforward consequence of the previous theorem, we have

Corollary 3.

A space X is an extremally disconnected P-space if and only if every continuous function from a dense subset S of X into a separable metric space $(Y,d)$ has a continuous extension to the whole X. In particular, X is an extremally disconnected P-space if and only if every dense subset S of X is C-embedded in X.

Recall that a cardinal $\kappa$ is measurable if there exists a measure of Ulam in a set X of cardinality $\kappa$. The class of all nonmeasurable cardinals is very extensive; in fact, it is closed under all the standard operations of cardinal arithmetic. Whether every cardinal is nonmeasurable is a celebrated unsolved problem (see [15] Ch. 12 for details).

Isbell proved in [17] that if the cardinal of an extremally disconnected P-space is nonmeasurable, then X is a discrete space. This result, together with the previous corollary shows

Corollary 4.

Every nondiscrete space X of nonmeasurable cardinality has a dense subset S which is not C-embedded in X.

It follows from the result of Isbell above that Theorem 2 is valid for every set-valued function whenever the measurable cardinals do not exist. If there is a measurable cardinal, Theorem 2 fails to be true for lower semicontinuous set-valued functions.

Example 2.

Let X be a discrete space. Suppose that the cardinal of X, say κ, is measurable and, consequently, X is not realcompact ([15] Theorem 14.29). Then $\upsilon \left(X\right)$ is an extremally disconnected P-space. Indeed, since X is a P-space, so is $\upsilon \left(X\right)$ ([15] Theorem 12.2). Moreover, being X extremally disconnected, $\beta \left(X\right)$ is also extremally disconnected ([15] Problem 6M.1), and since $\upsilon \left(X\right)$ is dense in $\beta \left(X\right)$, so is $\upsilon \left(X\right)$ ([15] Problem 1H.4). Similarly to Example 1, consider the lower semicontinuous set-valued function from $\upsilon \left(X\right)$ into $\mathcal{C}\left(\mathbb{R}\right)$ defined as

$$\phi \left(x\right)=\left\{\begin{array}{cc}\left\{1\right\},& x\in \upsilon \left(X\right)\setminus X,\hfill \\ \mathbb{I},& x\in X.\hfill \end{array}\right.$$

The function $f:X\to \mathbb{R}$ with $f\left(x\right)=0$ for all $x\in X$ is a selection for ${\phi |}_X$, which does not have a continuous extension to a selection for $\upsilon \left(X\right)$.

4. Conclusions

We have presented a new characterization of extremally disconnected spaces and extremally disconnected P-spaces by means of selection theory. Our results link two important classes of spaces with a powerful tool such as selection theory. In particular, our outcomes allow us to obtain the extension theorems for dense subsets of extremally disconnected spaces (respectively, extremally disconnected P-spaces).

It must be taken into account that extremally disconnected spaces appear in many fields of mathematics: Boolean algebras, Stone-Čech compactification of a discrete space, the spectrum of an abelian von Neumann algebra, the applications of the space ${\ell }_{\infty }$, the vector space of all bounded sequences of real numbers (in fact, ${\ell }_{\infty }$ can be identified with the vector space of all continuous functions on $\beta \left(\mathbb{N}\right)$), etc. Moreover, extremally disconnected P-spaces are related to the existence of measurable cardinals, an outstanding unsolved problem. Thus, the results presented in this paper permit the selection theory to be applied to various contexts of interest. In addition to these possible applications, future research can focus on the characterization of certain classes of topological spaces by using selection theory. In this framework, a characterization of realcompact spaces was obtained in [18], but such a characterization is unknown for interesting subclasses of realcompact spaces (for example, Lindelöf spaces).