Equalities for the extent of infinite products and \(\varSigma\)-products
For a topological cardinal function \(\varphi\), it is an interesting problem to ask when \(\varphi(X)= |\Lambda| \cdot \sup \{ \, \varphi (X_ \lambda ) \mid \lambda \in \Lambda \,\}\), for some product \(X = \prod _{ \lambda \in \Lambda} X_ \lambda\) of topological spaces [\textit{I. Juhász,} Cardinal functions in topology -- ten years later. Amsterdam: Mathematisch Centrum (1980; Zbl 0479.54001)]. Motivated by a comparison with known results about tightness, the authors study the above problem with respect to extent, as well as a similar problem concerning \(\Sigma\)-products. In detail, recall that the \emph{extent} of a topological space \(X\) is \[e(X) = \omega \cdot \sup \{ \, |D| \mid D \text{ is a closed discrete subset in } X \, \}. \] Compact factors in a product make no contribution to the extent of the product, hence all factors are supposed to be noncompact. All topological spaces are assumed to be regular and \(T_1\). For \(x \in X\) and a cover \(\mathcal U\) of \(X\), \( St(x, \mathcal U ) \) is \( \bigcup \{ \,U \in \mathcal U \mid x \in U \, \}\). A space \(X\) is a \emph{strict p-space} if there is a sequence \(\{ \mathcal U_n \}\) of open covers of \(X\) such that, for each \(x \in X\), both (i) \(\bigcap _{n \in \omega } St(x, \mathcal U _n)\) is compact, and (ii) for each open set \(U\) in \(X\) such that \(\bigcap _{n \in \omega } St(x, \mathcal U _n)\subset U\), there is an \( m \in \omega\) such that \( St(x, \mathcal U _m) \subset U\). A space \(X\) is a \emph{strong \(\Sigma\)-space} (resp., \emph{\(\Sigma\)-space}) if there are a sequence \(\{ \mathcal F_n \}\) of locally finite closed covers of \(X\) and a cover \(\mathcal K\) of \(X\) by compact (resp., countably compact closed) sets such that for each \(K \in \mathcal K\) and each open set \(U \in X\) with \(K \subset U\), there is an \(F \in \bigcup _{n \in \omega } \mathcal F_n\) with \(K \subset F \subset U\). Among many other interesting results, the following theorems are proved. \textbf{Theorem 1.} Assume that \( \Lambda\) is an infinite set and \(| \Lambda |\) is less than the first inaccessible cardinal (or there is no inaccessible cardinal). If \(X = \prod _{ \lambda \in \Lambda} X_ \lambda\) is a product of non-compact topological spaces and each factor is either a strict p-space or a strong-\(\Sigma\) space, then \(e(X)= |\Lambda| \cdot \sup \{ \, e (X_ \lambda ) \mid \lambda \in \Lambda \,\}\). \textbf{Theorem 2.} Let \(\Sigma\) be a \(\Sigma\)-product of the spaces \(X_ \lambda\) (\( \lambda \in \Lambda\)) and suppose that each \(X_ \lambda\) is either a strict p-space or a strong \(\Sigma\)-space. Then \(e(\Sigma)= \sup \{ \, e (X_ \lambda ) \mid \lambda \in \Lambda \,\}\). Counterexamples are provided showing that in the above theorems ``strong \(\Sigma\)-space'' cannot be weakened to ``\(\Sigma\)-space'' and that it is consistent with ZFC that ``strict p-space'' cannot be weakened to ``p-space''.
scientific article; zbMATH DE number 7464226
scientific article