Countably Compact Paratopological Groups

Abstract

By means of the theory of bispaces we show that a countably compact T0 paratopological group (G, τ) is a topological group if and only if (G, τ ∨ τ-1) is ω-bounded (here τ-1 is the conjugate topology of τ). Our approach is premised on the fact that every paratopological countably compact paratopological group is a Baire space and on the notion of a 2-pseudocompact space. We also prove that every ω-bounded (respectively, topologically periodic) Baire paratopological group admits a weaker Hausdorff group topology. In particular, ω-bounded (respectively, topologically periodic) 2-pseudocompact (so, also countably compact) paratopological groups enjoy this property. Some topological properties turning countably compact topological semigroups into topological groups are presented and some open questions are posed.