\(p\)-pseudocompactness and related topics in topological spaces (Q1807592)

Let \(p\) be a non-principal ultrafilter on \(\omega\). A subset \(Y\) of a space \(X\) is called \(p\)-bounded in \(X\) if every sequence \(\{V_{n}: n\in\omega\}\) of open sets in \(X\) with \(V_{n}\cap Y\neq\emptyset\), for each \(n\in\omega\), has a \(p\)-limit point \(x\in X\). The property of being \(p\)-bounded is productive and preserved by continuous functions. In addition, if \(X\) is \(p\)-bounded for some \(p\in\omega^{*}\), then \(X\) is pseudocompact [\textit{S. Garcia-Ferreira}, Ann. N.Y. Acad. Sci. 728, 22-31 (1994; Zbl 0911.54022)]. The authors establish some new properties of \(p\)-bounded subsets formulated in terms of \(z\)-ultrafilters and families of equicontinuous functions. Then they analyze the relations of \(p\)-pseudocompactness with \(p\)-compactness and \(\alpha\)-pseudocompactness, \(\alpha\geq\omega\). It is proved that \(cl_{\mu(X\times Y)}(A\times B)=cl_{\mu{X}}(A) \times cl_{\mu{Y}}(B)\) for any \(p\)-bounded subsets \(A\subseteq X\) and \(B\subseteq Y\), where \(\mu\) denotes the Dieudonné completion of a space. An analog of Glicksberg's pseudocompact product theorem is proved for \(p\)-boundedness.