Abstract
A Tychonoff space X is called \(\kappa \)-pseudocompact if for every continuous mapping f of X into \(\mathbb {R}^\kappa \) the image f(X) is compact. This notion generalizes pseudocompactness and gives a stratification of spaces lying between pseudocompact and compact spaces. It is well known that pseudocompactness of X is determined by the uniform structure of the function space \(C_p(X)\) of continuous real-valued functions on X endowed with the pointwise topology. In respect of that A.V. Arhangel’skii asked if analogous assertion is true for \(\kappa \)-pseudocompactness. We provide an affirmative answer to this question.
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1 Introduction
In this note, by a space we mean a Tychonoff topological space. For a space X, by \(C_p(X)\) we denote the space of all continuous real-valued functions on X endowed with the pointwise topology. The symbol \(C_p^*(X)\) stands for the subspace of \(C_p(X)\) consisting of all bounded continuous functions. Recall that X is pseudocompact if \(C_p(X)=C_p^*(X)\), i.e. every real-valued continuous function on X is bounded. In 1962 J.F. Kennison [7] introduced the following generalization of psudocompactness. Let \(\kappa \) be an infinite cardinal. A space X is called \(\kappa \)-pseudocompact if for every continuous mapping f of X into \(\mathbb {R}^\kappa \) the image f(X) is compact. Clearly, \(\kappa \)-pseudocompactness implies \(\lambda \)-pseudocompactness for every infinite cardinal \(\lambda \le \kappa \). Since metrizable pseudocompact spaces are compact, it is easy to see that \(\omega \)-pseudocompactness is precisely pseudocompactness. In particular any \(\kappa \)-pseudocompact space is pseudocompact. It was established by Uspenskiĭ in [12] that pseudocompactness of X is determined by the uniform structure of the function space \(C_p(X)\) (see [4] for a different proof of this result). In respect of that A.V. Arhangel’skii asked in 1998 if analogous result holds for \(\kappa \)-pseudocompactness (see [3, Question 13] or [11, Problem 4.4.2]).
The aim of the present note is to provide an affirmative answer to this question by proving the following extension of Uspenskiĭ’s theorem:
Theorem 1.1
For any infinite cardinal \(\kappa \), if \(C_p(X)\) and \(C_p(Y)\) are uniformly homeomorphic, then X is \(\kappa \)-pseudocompact if and only if Y is \(\kappa \)-pseudocompact.
Let us recall that that a map \(\varphi :C_p(X)\rightarrow C_p(Y)\) is uniformly continuous if for each open neighborhood U of the zero function in \(C_p(Y)\), there is and open neighborhood V of the zero function in \(C_p(X)\) such that \((f-g)\in V\) implies \((\varphi (f)-\varphi (g))\in U\). Spaces \(C_p(X)\) and \(C_p(Y)\) are uniformly homeomorphic if there is a homeomorphism \(\varphi \) between them such that both \(\varphi \) and \(\varphi ^{-1}\) are uniformly continuous.
The proof of Theorem 1.1 is inspired by author’s recent work [8] concerned with linear homeomorphisms of function spaces. The basic idea in [8] relies on the fact that certain topological properties of a space X can be conveniently characterized by the way X is positioned in its Čech-Stone compactification \(\beta X\); \(\kappa \)-pseudocompactess is one of such properties. Indeed, Hewitt [6] gave the following description of pseudocompactness (cf. [1, Theorem 1.3.3]).
Theorem 1.2
(Hewitt). A space X is pseudocompact if and only if every nonempty \(G_\delta \)-subset of \(\beta X\) meets X.
It was noted by Retta in [10] that the above result easily extends to \(\kappa \)-pseudocompactness. We need the following notation. Let \(\kappa \) be an infinite cardinal. A subset A of a space Z is a \(G_\kappa \)-set if it is an intersection of at most \(\kappa \)-many open subsets of Z. The \(G_{\omega }\)-sets are called \(G_\delta \)-sets and the complement of a \(G_\delta \)-set is called \(F_\sigma \)-set. We have (see [10, Theorem 1]):
Theorem 1.3
(Retta). Let \(\kappa \) be an infinite cardinal. A space X is \(\kappa \)-pseudocompact if and only if every nonempty \(G_\kappa \)-subset of \(\beta X\) meets X.
The uniform structure of spaces of continuous functions was studied by many authors; the interested reader should consult the book [11]. For our purposes, the most important are some ideas developed by Gul’ko in [5].
2 Results
For a space Z and a function \(f\in C^*_p(Z)\) the function \({\widetilde{f}}:\beta Z\rightarrow \mathbb {R}\) is the unique continuous extension of f over the Čech-Stone compactification \(\beta Z\) of Z. Let \(\varphi :C^*_p(X)\rightarrow C^*_p(Y)\) be a uniformly continuous surjection. For \(y\in \beta Y\) and a subset K of \(\beta X\) we define
Note that \(a(y,\emptyset )=\infty \) since \(\varphi \) is onto.
For \(y\in \beta Y\) define the family
Similarly, for \(y\in \beta Y\) and \(n\in \mathbb {N}\) let
It may happen that for some n the family \(\mathscr {A}_n(y)\) is empty. However, we have the following:
Proposition 2.1
For every \(y\in Y\), there exists n for which \(\mathscr {A}_n(y)\) contains a nonempty finite subset of X. In particular, for this n the family \(\mathscr {A}_n(y)\) is nonempty.
Proof
By uniform continuity of \(\varphi \), there is \(n\in \mathbb {N}\) and a finite subset F of X such that
We claim that \(F\in \mathscr {A}_n(y)\). To see this, take arbitrary functions \(f,g\in C_p^*(X)\) such that \(|f(x)-g(x)|<1\), for every \(x\in F\). Put \(f_k=f+\tfrac{k}{n}(g-f)\), for \(k=0,1,\ldots , n\). Then \(f_0=f\), \(f_n=g\) and \(|f_k(x)-f_{k+1}(x)|<1/n\) for \(x\in F\). Hence, by (1) we get
as required. \(\square \)
Clearly, for every \(y\in \beta Y\) we have \(\mathscr {A}(y)=\bigcup _{n\in \mathbb {N}}\mathscr {A}_n(y)\). In particular, for \(y\in Y\) the family \(\mathscr {A}(y)\) is always nonempty.
For \(n\in \mathbb {N}\) we set
Note that \(y\in Y_n\) if and only if \(\beta X\in \mathscr {A}_n(y)\). Using this observation it is easy to show the following:
Lemma 2.2
For every \(n\in \mathbb {N}\) the set \(Y_n\) is closed in \(\beta Y\); hence compact.
Proof
Pick \(y\in \beta Y\setminus Y_n\). Since \(\beta X\notin \mathscr {A}_n(y)\), there are functions \(f,g\in C_p^*(X)\) satisfying \(|{\widetilde{f}}(x)-{\widetilde{g}}(x)|<1\) for every \(x\in \beta X\), and \(|\widetilde{\varphi (f)}(y)-\widetilde{\varphi (g)}(y)|>n\). The set
is an open neighborhood of y in \(\beta Y\). Moreover if \(z\in U\), then f and g witness that \(\beta X\notin \mathscr {A}_n(z)\); thus \(U\cap Y_n=\emptyset \). \(\square \)
For a space X and a positive integer m, we denote by \([X]^{\le m}\) the space of all nonempty at most m-element subsets of X endowed with the Vietoris topology, i.e. basic open sets in \([X]^{\le m}\) are of the form
where \(\{U_1,\ldots ,U_k\}\) is a finite collection of open subset of X.
For any positive integers n, m we define
Note that \(Y_{n,m}\subseteq Y_n\) and by Proposition 2.1 we have
We claim that \(Y_{n,m}\) is closed in \(Y_n\) and hence it is compact:
Lemma 2.3
The set \(Y_{n,m}\) is closed in \(Y_n\), hence it is compact.
Proof
Consider the following subset Z of the product \(Y_n\times [\beta X]^{\le m}\)
We show that Z is closed. Pick \((y,F)\in (Y_n\times [\beta X]^{\le m}){\setminus } Z\). Then \(F\notin \mathscr {A}_n(y)\) and thus there are \(f,g\in C_p^*(X)\) satisfying \(|{\widetilde{f}}(x)-{\widetilde{g}}(x)|<1\) for every \(x\in F\), and \(|\widetilde{\varphi (f)}(y)-\widetilde{\varphi (g)}(y)|>n\). Let \(U=\{x\in \beta X:|{\widetilde{f}}(x)-{\widetilde{g}}(x)|<1\}\) and \(V=\{z\in Y_n:|\widetilde{\varphi (f)}(z)-\widetilde{\varphi (g)}(z)|>n\}\). The set \(V\times \langle U \rangle \) is an open neighborhood of (y, F) in \(Y_n\times [\beta X]^{\le m}\) disjoint from Z.
The set Z, being is closed in the compact space \(Y_n\times [\beta X]^{\le m}\), is compact. Since the set \(Y_{n,m}\) is the image of Z under the projection map it must be compact. \(\square \)
Corollary 2.4
Suppose that Y is pseudocompact and let \(\varphi :C_p^*(X)\rightarrow C_p^*(Y)\) be a uniformly continuous surjection. For every \(y\in \beta Y\), there exist n and m such that \(y\in Y_{n,m}\).
Proof
By Lemma 2.3, the set \(\bigcup _{n,m} Y_{n,m}\) is \(F_\sigma \) in \(\beta Y\) and contains Y, by (2). It follows from Theorem 1.2 that \(\bigcup _{n,m}Y_{n,m}=\beta Y\). \(\square \)
For \(y\in \bigcup _{n,m} Y_{n,m}\) we define
Remark
For \(y\in Y\) the set K(y) is the support introduced by Gul’ko in [5] (see also [2, 4, 9]).
Lemma 2.5
For every \(y\in \bigcup _{n,m} Y_{n,m}\) the set K(y) is a nonempty finite subset of \(\beta X\). Moreover, \(K(y)\in \mathscr {A}(y)\). If \(y\in Y\), then K(y) is a subset of X.
Proof
We show that the family \(\mathscr {A}(y)\) is closed under finite intersections. Pick \(K_1,K_2\in \mathscr {A}(y)\) and let \(f,g\in C_p^*(X)\) be such that \(|{\widetilde{f}}(x)-{\widetilde{g}}(x)|<1\) for every \(x\in K_1\cap K_2\). Let
The set U is open in \(\beta X\) and \(K_1\cap K_2\subseteq U\).
Since \(K_1\) and \(K_2{\setminus } U\) are disjoint closed subsets of the compact space \(\beta X\), by Urysohn’s lemma there is a continuous function \(u:\beta X\rightarrow [0,1]\) such that
Let
and let \(h\in C^*_p(X)\) be the restriction of \({\widetilde{h}}\) to X. We have:
-
\({\widetilde{h}}(x)={\widetilde{f}}(x)\) for \(x\in K_1\),
-
\({\widetilde{h}}(x)={\widetilde{g}}(x)\) for \(x\in K_2{\setminus } U\) and
-
if \(x\in U\), then \(|{\widetilde{h}}(x)-{\widetilde{g}}(x)|=|u(x)|\cdot |{\widetilde{f}}(x)-{\widetilde{g}}(x)|<1\), by definition of U and the fact that u maps into [0, 1].
In particular, since \(K_1\cap K_2\subseteq U\), we get
-
\(|{\widetilde{h}}(x)-{\widetilde{g}}(x)|<1\) for \(x\in K_2\).
Since \(K_1\in \mathscr {A}(y)\) and \({\widetilde{h}}(x)={\widetilde{f}}(x)\) for \(x\in K_1\), we get \(|\widetilde{\varphi (f)}(y)-\widetilde{\varphi (h)}(y)|\le a(y,K_1)<\infty \). Similarly, since \(|{\widetilde{h}}(x)-{\widetilde{g}}(x)|<1\) for \(x\in K_2\) and \(K_2\in \mathscr {A}(y)\), we have \(|\widetilde{\varphi (g)}(y)-\widetilde{\varphi (h)}(y)|\le a(y,K_2)<\infty \). Hence,
So \(a(y,K_1\cap K_2)\le a(y,K_1)+a(y,K_2)\) and thus \(K_1\cap K_2\in \mathscr {A}(y)\). By induction the result follows for any finite intersection.
The family \(\mathscr {A}(y)\), consisting of closed subsets of \(\beta X\), is closed under finite intersections and \(\emptyset \notin \mathscr {A}(y)\) so by compactness the intersection \(\bigcap \mathscr {A}(y)\) must be nonempty. It is finite because \(y\in Y_{n,m}\) guarantees that the family \(\mathscr {A}(y)\) contains a subset of \(\beta X\) which is at most m-element.
Since \(\mathscr {A}(y)\) contains a finite subset F of \(\beta X\), the set \(K(y)\subseteq F\) is an intersection of finitely many elements of \(\mathscr {A}(y)\) so the first part of the proof implies that \(K(y)\in \mathscr {A}(y)\).
Finally, if \(y\in Y\) then \(y\in \bigcup _{n,m}Y_{n,m}\), by (2). So K(y) is well defined. The inclusion \(K(y)\subseteq X\) follows from Proposition 2.1. \(\square \)
For \(y\in \bigcup _{n,m} Y_{n,m}\) we define
By Lemma 2.5, \(K(y)\in \mathscr {A}(y)\) so \(a(y)<\infty \). For a subset A of \(\beta X\) we set
Combining Corollary 2.4 and Lemma 2.5 we get
Proposition 2.6
Suppose that \(\varphi :C^*_p(X)\rightarrow C^*_p(Y)\) is a uniformly continuous surjection. If Y is pseudocompact then, for every \(y\in \beta Y\), the set K(y) is a well-defined nonempty finite subset of \(\beta X\) that belongs to the family \(\mathscr {A}(y)\). Also, a(y) is a well-defined number, for every \(y\in \beta Y\).
The proof of the next lemma is analogous to the proof of [2, Lemma 1.3].
Lemma 2.7
Suppose that \(U\subseteq \beta X\) is open and let n be a positive integer. For every \(y\in K^{-1}(U)\cap Y_n\) there exists an open neighborhood V of y in \(\beta Y\) such that for every \(z\in V\cap Y_n\) and every \(A\in \mathscr {A}_n(z)\) we have \(A\cap U\ne \emptyset \).
Proof
Fix \(x_0\in K(y)\cap U\) witnessing \(y\in K^{-1}(U)\). Since K(y) is finite, shrinking U if necessary we can assume that \(U\cap K(y)=\{x_0\}\). Note that \(\beta X{\setminus } U\notin \mathscr {A}(y)\) for otherwise K(y) would be a subset of \(\beta X\setminus U\) and this is not the case because \(x_0\in K(y)\cap U\). It follows that there are \(f,g\in C^*_p(X)\) such that
Let \({\widetilde{h}}\in C_p(\beta X)\) be a function satisfying
and let \(h\in C^*_p(X)\) be the restriction of \({\widetilde{h}}\).
Note that by (3) and (5), \(|{\widetilde{h}}(x)-{\widetilde{g}}(x)|<1\) for every \(x\in K(y)\). Therefore,
According to (4) and (6) we have
Let
The set V is open and \(y\in V\), by (7). We show that V is as required.
Take \(z\in V\cap Y_n\) and let \(A\in \mathscr {A}_n(z)\). If \(A\cap U=\emptyset \) then \({\widetilde{h}}(x)={\widetilde{f}}(x)\) for every \(x\in A\), by (5). So \(|\widetilde{\varphi (f)}(z)-\widetilde{\varphi (h)}(z)|\le n\), contradicting \(z\in V\). \(\square \)
Proposition 2.8
Suppose that Y is pseudocompact. If \(U\subseteq \beta X\) is open, then the set \(K^{-1}(U)\) is a \(G_\delta \)-subset of \(\beta Y\).
Proof
By Proposition 2.6, for every \(y\in \beta Y\), the set K(y) is a nonempty finite subset of \(\beta X\). For \(n=1,2,\ldots \), let
For \(y\in L_n\) let \(V_n^y\) be an open neighborhood of y in \(\beta Y\) provided by Lemma 2.7, i.e.
Let
We claim that
Indeed, pick \(y\in K^{-1}(U)\) and fix an arbitrary \(m\ge 1\). Since \(\beta Y=\bigcup _{n=1}^\infty Y_n\) (cf. Corollary 2.4), there is i such that \(y\in Y_i\). Since \(Y_n\subseteq Y_{n+1}\), we can assume that \(i>m\). We have \(y\in L_i\) whence \(y\in V^y_i\subseteq V_i\subseteq \bigcup _{n=m}^\infty V_n\), because \(i>m\).
To prove the opposite inclusion, take \(z\in \bigcap _{m=1}^\infty \bigcup _{n=m}^\infty V_n\). Again, there is i such that \(z\in Y_i\). Let j be a positive integer satisfying \(j>\max \{a(z),i\}\). By our assumption, \(z\in \bigcup _{n=j}^\infty V_n\), so there is \(k\ge j\) such that \(z\in V_k\). Clearly, \(z\in Y_k\) and since \(k>a(z)\) we have
By definition of \(V_k\), there is \(y\in L_k\) such that \(z\in V^y_k\). Now, from (8) and (9) we get \(z\in K^{-1}(U)\). \(\square \)
Remark 2.9
If \(\varphi :C^*_p(X)\rightarrow C^*_p(Y)\) is a uniform homeomorphism, we may consider the inverse map \(\varphi ^{-1}:C^*_p(Y)\rightarrow C^*_p(X)\) and apply all of the above results to \(\varphi ^{-1}\). In particular, if X is pseudocompact, then for every \(x\in \beta X\) we can define the set \(K(x)\subseteq \beta Y\) and the real number a(x) simply by interchanging the roles of X and Y above.
Lemma 2.10
Suppose that both X and Y are pseudocompact spaces. Let \(\varphi :C^*_p(X)\rightarrow C^*_p(Y)\) be a uniform homeomorphism. For any \(x\in \beta X\) there is \(y\in K(x)\) such that \(x\in K(y)\).
Proof
Let \(x\in \beta X\). Applying Proposition 2.6, first to \(\varphi ^{-1}\) and then to \(\varphi \), we infer that the set \(F=\bigcup \{K(y):y\in K(x)\}\subseteq \beta X\) is finite being a finite union of finite sets. Let M be a positive integer such that
Striving for a contradiction, suppose that \(x\notin F\). Let \({\widetilde{f}},{\widetilde{g}}\in C_p(\beta X)\) be functions satisfying
Let \(f\in C^*_p(X)\) and \(g\in C^*_p(X)\) be the restrictions of \({\widetilde{f}}\) and \({\widetilde{g}}\), respectively. Since for every \(y\in K(x)\) the functions \({\widetilde{f}}\) and \({\widetilde{g}}\) agree on \(K(y)\subseteq F\), we have
For \(k\in \{0,1,\ldots , M\}\) define a function \(\widetilde{h_k}\in C_p(\beta Y)\) by the formula
Obviously, \(\widetilde{h_0}=\widetilde{\varphi (f)}\) and \(\widetilde{h_M}=\widetilde{\varphi (g)}\). Moreover, by (11), we have
For \(k\in \{0,1,\ldots , M\}\) let \(h_k\in C^*_p(Y)\) be the restriction of \(\widetilde{h_k}\). Using (12) we get:
This however contradicts (10). \(\square \)
Now we are ready to prove of our main result.
Proof of Theorem 1.1
Let \(\kappa \) be an infinite cardinal and let \(\varphi :C_p(X)\rightarrow C_p(Y)\) be a uniform homeomorphism. By symmetry it is enough to show that if Y is \(\kappa \)-pseudocompact then so is X. So let us assume that Y is \(\kappa \)-pseudocompact. Then, in particular Y is pseudocompact and hence by Uspenskiĭ’s theorem [12, Corollary] (cf. [11, V.136]), so is X. Hence, \(C_p(Y)=C^*_p(Y)\) and \(C_p(X)=C^*_p(X)\). In order to prove that X is \(\kappa \)-pseudocompact we will employ Theorem 1.3. For this purpose fix a nonempty \(G_\kappa \)-subset G of \(\beta X\). It suffices to prove that \(G\cap X\ne \emptyset \).
Claim 1
The set \(K^{-1}(G)=\{y\in \beta Y:K(y)\cap G\ne \emptyset \}\) is nonempty.
Proof
The set G is nonempty so let us fix \(x\in G\). According to Lemma 2.10 there is \(y\in K(x)\) such that \(x\in K(y)\). In particular, \(y\in K^{-1}(G)\). \(\square \)
Claim 2
The set \(K^{-1}(G)=\{y\in \beta Y:K(y)\cap G \ne \emptyset \}\) is a \(G_\kappa \)-set in \(\beta Y\).
Proof
Write \(G=\bigcap \{U_\alpha :\alpha <\kappa \}\), where each \(U_\alpha \) is an open subset of \(\beta X\). We can also assume that the family \(\{U_\alpha :\alpha <\kappa \}\) is closed under finite intersections. It follows from Proposition 2.8 that for each \(\alpha <\kappa \), the set \(K^{-1}(U_\alpha )\) is \(G_\delta \) in \(\beta Y\). Thus, it is enough to show that
To this end, take \(y\in \bigcap _{\alpha < \kappa }K^{-1}(U_\alpha )\). According to Proposition 2.6, the set K(y) is a nonempty finite subset of \(\beta X\). Enumerate \(K(y)=\{x_1,\ldots ,x_k\}\), where k is a positive integer. If \(y\notin K^{-1}(G)\), then for every \(i\le k\) there is \(\alpha _i<\kappa \) such that
The family \(\{U_\alpha :\alpha <\kappa \}\) is closed under finite intersections, so there is \(\gamma <\kappa \) with \(U_\gamma =U_{\alpha _1}\cap \cdots \cap U_{\alpha _k}\). But \(y\in \bigcap _{\alpha < \kappa }K^{-1}(U_\alpha )\subseteq K^{-1}(U_\gamma )\). Hence, there is \(j\le k\) such that \(x_j\in U_\gamma \subseteq U_{\alpha _j}\), which is a contradiction with (13). Therefore, we must have \(y\in K^{-1}(G)\). This provides the inclusion \(K^{-1}(G)\supseteq \bigcap _{\alpha < \kappa }K^{-1}(U_\alpha )\). The opposite inclusion is immediate. \(\square \)
It follows from Claims 1 and 2 that the \(K^{-1}(G)\) is a nonempty \(G_\kappa \)-subset of \(\beta Y\). Hence, by Theorem 1.3, there exists \(p\in K^{-1}(G)\cap Y\). We have \(K(p)\cap G\ne \emptyset \) and since \(p\in Y\), we infer from Lemma 2.5 that K(p) is a nonempty finite subset of X. Therefore, \(\emptyset \ne K(p)\cap G\subseteq X\cap G\). \(\square \)
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Krupski, M. On \(\kappa \)-Pseudocompactess and Uniform Homeomorphisms of Function Spaces. Results Math 78, 154 (2023). https://doi.org/10.1007/s00025-023-01932-4
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DOI: https://doi.org/10.1007/s00025-023-01932-4
Keywords
- Function space
- pointwise convergence topology
- \(C_p(X)\) space
- u-equivalence
- uniform homeomorphism
- \(\kappa \)-pseudocompactness