Continuous mappings on subspaces of products with the \( \kappa \)-box topology
The authors prove that under suitable assumptions, a continuous function \(f\) on a product space \(X\) endowed with a certain box topology depends only on a ``few'' coordinates, and study the existence of continuous extensions of \(f\) to spaces \(Y\) containing \(X\). The obtained results generalize some results of \textit{W.\ W.\ Comfort} and \textit{S.\ Negrepontis} [Chain conditions in topology, Cambridge Tracts in Mathematics 79 (2008; Zbl 1156.54002)] and \textit{M.\ Hušek} [Symp.\ math.\ 17, Anelli Funz.\ contin., Gruppi infin., Convegni 1973, 25--41 (1976; Zbl 0372.54005)]. If \(\kappa\) and \(\alpha\) are cardinals, we write \(\kappa \ll \alpha\) if \(\kappa < \alpha\), and for every \(\beta < \alpha\) and \(\lambda< \kappa\) we have \(\beta ^\lambda < \alpha\). Let \(X\) be a topological space. A family \(\{ A_i : i \in I\}\) of non-empty subsets of \(X\) is \textit{locally \(< \kappa\)} if there is an open cover \({\mathcal U}\) of \(X\) such that \(|\{i \in I : U \cap A_i \neq \emptyset\}| < \kappa\) for each \(U \in {\mathcal U}\). The space \(X\) is \textit{pseudo-\((\alpha, \kappa )\)-compact} if every locally \(< \kappa\) family \(\{U _i : i \in I \}\) of non-empty open sets satisfies \(| I | < \alpha\). For every \(\{ X_i : i \in I\}\) and \(J \subseteq I\) we write \(X_J = \prod _{i \in J} X_i\). When each \(X_i\) is a topological space, \((X_I)_{\kappa }\) denotes \(X_I\) endowed with the \(\kappa\)-box topology, i.e.\ for which \(\{\prod _{i \in I} U_i : U_i \subseteq X_{i} ~(i \in I)\text{ are open}, |\{i \in I : U_{i} \neq X_{i}\}| < \kappa \}\) is a base. If \(Y\) and \(Z\) are topological spaces, we denote by \(C (Y , Z )\) the set of continuous functions from \(Y\) into \(Z\), and \(\Delta_{Z} = \{(z,z) : z \in Z\} \subseteq Z \times Z\). We cite three representative results from the paper. Theorem. Let \(\omega \leq \kappa \leq \alpha\) be cardinals such that either \(\kappa < \alpha\) or \(\alpha\) is regular. Let \(\{ X_i : i \in I\}\) be a set of non-empty spaces, \(Y\) be a pseudo-\((\alpha, \kappa )\)-compact subspace of \((X_I )_\kappa\) which is dense in some open subset of \((X_I )_\kappa\). Let \(Z\) be an arbitrary space. Then for every open neighborhood \(O\) of \(\Delta_{Z}\) in \(Z \times Z\) and for every \(f \in C (Y , Z )\), there is \(J \in [ I ]^{ <\alpha}\) such that \(( f (x), f ( y)) \in \overline O\) whenever \(x, y \in Y\) are such that \(x_J = y_J\). Theorem. Suppose \(\omega \leq \kappa \ll \text{cf}(\alpha)\), \(\kappa \ll \alpha\), \(\kappa \leq \beta \leq \alpha\). Let \(\{ X_i : i \in I \}\) be a set of non-empty spaces and let \(Y \subseteq (X_I )_\kappa\). Assume that {\parindent4mm \begin{itemize}\item[--] either \(\alpha\) is regular and for every non-empty \(J \in [ I ] ^{<\kappa}\), \(\pi_{J} [Y ] = X_J\) and \((X_J )_\kappa\) is pseudo-\((\alpha, \beta )\)-compact; \item[--] or \(\alpha\) is singular and for every non-empty \(J \in [ I ] ^{\leq \text{cf}(\alpha)}\), \(\pi_{J} [Y ] = X_J\) and \((X_J )_\kappa\) is pseudo-\((\alpha, \beta )\)-compact. \end{itemize}} Then \(Y\) is pseudo-\((\alpha, \beta )\)-compact. Theorem. Let \(\omega \leq \kappa \leq \alpha\) be cardinals such that either \(\kappa < \alpha\) or \(\alpha\) is regular, and let \( \omega \leq \beta\). Let \(\{ X_i : i \in I \}\) be a set of non-empty spaces, and let \(Y\) be a pseudo-\((\alpha, \kappa )\)-compact subspace of \((X_I )_\kappa\). Assume also that {\parindent4mm \begin{itemize}\item[--] either \(\pi_J [Y ] = X_J\) for every non-empty \(J \in [ I ] ^{\leq \alpha \cdot \beta}\); \item[--] or \(\beta < \text{cf}(\alpha)\), and \(\pi_J [Y ] = X_J\) for every non-empty \(J \in [ I ] ^{<\alpha}\). \end{itemize}} Let \(Z\) be a space for which \(\Delta_{Z}\) is the intersection of \(\beta\) many regular-closed subsets of \(Z \times Z\). Then every \(f \in C (Y , Z )\) extends to a continuous function \(\overline f : (X_I )_\kappa \rightarrow Z\).
scientific article; zbMATH DE number 5625458
scientific article