On the product of a compact space with an absolutely countably compact space.

*(English)*Zbl 0917.54023
Coplakova, Eva (ed.) et al., Papers on general topology and applications. Proceedings of the 10th summer conference, Amsterdam, Netherlands, August 15–18, 1994. New York, NY: New York Academy of Sciences. Ann. N. Y. Acad. Sci. 788, 203-208 (1996).

It is known that a space \(X\) is countably compact if and only if for every open cover \(\mathcal U\), there is a finite subset \(F\) of \(X\) such that \(St(F,\mathcal U)=X\). A space \(X\) is absolutely countably compact (acc) provided that for every open cover \(\mathcal U\) and dense subset \(D\), there is a finite subset \(F\) of \(D\) such that \(St(F,\mathcal U)=X\). Unlike countable compactness, the stronger acc property of \(X\) is not preserved by taking a product of \(X\) with a compact space. In this paper, sufficient conditions are given for such a product with a compact space to be acc. The main result is that if \(Y\) is a compact sequential \(T_2\)-space and \(X\) is an acc \(T_3\)-space then \(X\times Y\) is acc. It also follows from a related result that every countably compact GO-space is acc.

For the entire collection see [Zbl 0903.00049].

For the entire collection see [Zbl 0903.00049].

Reviewer: R.A.McCoy (Blacksburg)