Pseudocompactness, Products, and Topological Brandt λ0 -Extensions of Semitopological Monoids

Resumo

In the present paper, we study the preservation of pseudocompactness (resp., countable compactness, sequential compactness, ω -boundedness, totally countable compactness, countable pracompactness, sequential pseudocompactness) by Tychonoff products of pseudocompact (and countably compact) topological Brandt \( {\lambda}_i^0 \) -extensions of semitopological monoids with zero. In particular, we show that if \( \left\{\left({B}_{\uplambda_i}^0\left({S}_i\right),\kern0.5em {\uptau}_{B\left({S}_i\right)}^0\right):i\in \mathrm{\mathcal{I}}\right\} \) is a family of Hausdorff pseudocompact topological Brandt \( {\uplambda}_i^0 \) -extensions of pseudocompact semitopological monoids with zero such that the Tychonoff product \( \prod \left\{{S}_i:i\in \mathrm{\mathcal{I}}\right\} \) is a pseudocompact space, then the direct product \( \prod \left\{\left({B}_{\uplambda_i}^0\left({S}_i\right),\kern0.5em {\uptau}_{B\left({S}_i\right)}^0\right):i\in \mathrm{\mathcal{I}}\right\} \) endowed with the Tychonoff topology is a Hausdorff pseudocompact semitopological semigroup.