Pseudocompact paratopological groups that are topological

Abstract

We obtain necessary and sufficient conditions when a pseudocompact paratopological group is topological. (2-)pseudocompact and countably compact paratopological groups that are not topological are constructed. It is proved that each 2-pseudocompact paratopological group is pseudocompact and that each Hausdorff σ-compact pseudocompact paratopological group is a compact topological group. Our particular attention is devoted to periodic and topologically periodic pseudocompact paratopological groups.

On feebly compact inverse primitive (semi)topological semigroups

“…Since by Proposition 3 of [30] inversion on a quasiregular feebly compact paratopological group is continuous, Proposition 3.14, Theorems 3.20 and 3.22 imply the following corollary: Corollary 3.23. Inversion on a quasi-regular primitive inverse feebly compact topological semigroup S is continuous and hence S is Tychonoff.…”

“…Then every non-zero maximal subgroup of S is a quasi-regular space and hence by Proposition 3 of [30] (see also [31]) every maximal subgroup of S is a topological group. Now, Proposition 2.5 of [17] implies that…”