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

TL;DR

This paper investigates how various forms of compactness are preserved in Tychonoff products of topological Brandt $oldsymbol{ ext{λ}^0}$-extensions of semitopological monoids, establishing conditions for pseudocompactness preservation.

Contribution

It provides new results on the preservation of pseudocompactness in products of topological Brandt extensions of semitopological monoids with zero.

Findings

Product of pseudocompact semitopological monoids with zero is pseudocompact.

Tychonoff product of Hausdorff pseudocompact Brandt extensions remains pseudocompact.

Conditions under which pseudocompactness is preserved in product spaces.

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 $\big\{ \big(B^0_{\lambda_i}(S_i),\tau^0_{B(S_i)}\big) \colon i\in\mathscr{I}\big\}$ is a family of Hausdorff pseudocompact to\-pological Brandt $\lambda_i^0$-extensions of pseudocompact semitopological monoids with zero such that the Tychonoff product $\prod\left\{ S_i \colon i\in\mathscr{I}\right\}$ is a pseudocompact space then the direct product $\prod\big\{ \big(B^0_{\lambda_i}(S_i),\tau^0_{B(S_i)}\big) \colon i\in\mathscr{I}\big\}$ endowed with the Tychonoff topology is a Hausdorff pseudocompact semitopological semigroup.