For Hausdorff spaces, \(H\)-closed = \(D\)-pseudocompact for every ultrafilter \(D\) (Q2828643)

We recall three definitions, with no separation axioms assumed in this article, where the first one is similar to one due to \textit{Z. Frolík} [Czech. Math. J. 9(84), 172--217 (1959; Zbl 0098.14201)], the second one was given by \textit{C. T. Scarborough} and \textit{A. H. Stone} [Trans. Am. Math. Soc. 124, 131--147 (1966; Zbl 0151.30001)], and the third one was given for Tychonoff spaces by \textit{J. Ginsburg} and \textit{V. Saks} [Pac. J. Math. 57, 403--418 (1975; Zbl 0288.54020)].NEWLINENEWLINEFor an infinite cardinal number \(\lambda\), a topological space is called \textit{weakly initially} \(\lambda\)-\textit{compact} if every open cover of cardinality \(\leq\lambda\) has a finite subfamily with dense union. A space is called \(H(i)\) if every open cover has a finite subfamily with dense union (a property which is equivalent to \(H\)-closedness for Hausdorff spaces). If \(D\) is an ultrafilter over a set \(I\), a space \(X\) is said to be \(D\)-\textit{pseudocompact} provided that for every indexed family of nonempty open sets \(\{O_i:i\in I \}\) of \(X\), there exists \(x\in X\) such that for every neighborhood \(U\) of \(x\), \(\{i\in I:U\cap O_i\neq\emptyset\}\in D\).NEWLINENEWLINEThe author obtains a number of theorems concerning these concepts, such as the following. If \(X\) is a weakly initially \(\lambda\)-compact topological space, and \(2^{\mu}\leq\lambda\), then \(X\) is \(D\)-pseudocompact for every ultrafilter \(D\) over any set of cardinality \(\leq\mu\). One corollary to this theorem is that if \(2^{\mu}\leq\lambda\), then the product of any family of weakly initially \(\lambda\)-compact spaces is weakly initially \(\mu\)-compact.NEWLINENEWLINEAnother theorem is that for a topological space \(X\) the following are equivalent: \(X\) is \(H(i)\); \(X\) is weakly initially \(\lambda\)-compact for every infinite cardinal number \(\lambda\); \(X\) is \(D\)-pseudocompact for every ultrafilter \(D\); and for every infinite cardinal number \(\lambda\), there is some regular ultrafilter \(D\) over \(\lambda\) such that \(X\) is \(D\)-pseudocompact (the term \textit{regular over} \(\lambda\) means there is a family of \(\lambda\) elements of \(D\) such that the intersection of any infinite subset of the family is empty). Several other results are also presented.