Pseudocompactness, products and topological Brandt ${\lambda }^0$-extensions of semitopological monoids

Abstract

In the paper we study the preservation of pseudocompactness (resp., countable compactness, sequential compactness, $\omega$-boundedness, totally countable compactness, countable pracompactness, sequential pseudocompactness) by Tychonoff products of pseudocompact (and countably compact) to\-pological Brandt ${\lambda }_i^0$-extensions of semitopological monoids with zero. In particular we show that if $\left\{\right(B_{{\lambda }_i}^0(S_i),{\tau }_{B(S_i)}^0):i\in \mathcal{I}\}$ is a family of Hausdorff pseudocompact to\-pological Brandt ${\lambda }_i^0$-extensions of pseudocompact semitopological monoids with zero such that the Tychonoff product $\prod \left\{S_i:i\in \mathcal{I}\right\}$ is a pseudocompact space then the direct product $\prod \left\{\right(B_{{\lambda }_i}^0(S_i),{\tau }_{B(S_i)}^0):i\in \mathcal{I}\}$ endowed with the Tychonoff topology is a Hausdorff pseudocompact semitopological semigroup.

arXiv admin note: substantial text overlap with arXiv:1310.4313