On properties similar to pseudocompactness and countable compactness
A topological space X is weakly countably compact if there is a dense set A in X such that every infinite subset of A has an accumulation point in X; for a natural number k, X is k-pseudocompact if for every open cover \({\mathcal U}\) of X one can find a finite subcover of X from \(\{St^ k(x,{\mathcal U});x\in X\}\). It is shown that 1-pseudocompactness coincides with countable compactness in regular spaces, k-pseudocompactness for \(k\geq 3\) coincides with pseudocompactness in completely regular spaces, every weakly countably compact space is 2-pseudocompact. It is also proved that weak countable compactness is preserved by open-closed weakly countably compact surjections, that product of two weakly countably compact spaces has the same property provided one factor is a k-space and that a regular weakly countably compact space is compact metrizable provided it has a point-countable base. The last result generalizes the result of \textit{A. Mishchenko} [Dokl. Akad. Nauk SSSR 144, 985-988 (1962; Zbl 0122.173)] but not its generalization by \textit{A. V. Arkhangelskij} [Usp. Mat. Nauk 33, No.6(204), 29-84 (1978; Zbl 0414.54002)] since one cannot use a pseudo base instead of a base in this case.
scientific article; zbMATH DE number 3886727