Pseudo-compactness, products and topological Brandt \(\lambda^0\)-extensions of semitopological monoids (Q2812885)

In the paper the authors discuss preservation of pseudo-compactness (countable compactness, sequential compactness, \(\omega\)-boundedness, totally countable compactness, countable pre-compactness, sequential pseudo-compactness) by the Tychonoff products of pseudo-compact (and countably compact) topological Brandt \(\lambda^0_i\)-extensions of semi-topological monoids with zero. In particular, it is shown that if \(\{(B^0_{\lambda_i}(S_i), \tau^0_{B(S_i)})\colon i\in\mathcal I\}\) is a family of Hausdorff pseudo-compact topological Brandt \(\lambda^0_i\)-extensions of pseudo-compact semi-topological monoids with zero such that the Tychonoff product \(\prod\{S_i\colon i\in\mathcal I\}\) is a pseudo-compact space, then the direct product \(\prod\{(B^0_{\lambda_i}(S_i), \tau^0_{B(S_i)})\colon i\in\mathcal I\}\) endowed with the Tychonoff topology is a Hausdorff pseudo-compact semi-topological semigroup.