Almost paratopological groups

Abstract

A class of almost paratopological groups is introduced, which (1) contains paratopological groups and Hausdorff quasitopological groups; (2) is closed under products; (3) subgroups. Almost paratopological T1 groups G are characterized by the fact that {(x,y)∈G2:xy=e} is closed in G2. A compact almost paratopological group is topological. A regular Σ-space with countable extend and a separately continuous Mal'tsev operation is ω-cellular (and ccc). A σ-compact regular almost paratopological group is ccc. In particular, a σ-compact regular quasitopological group is ccc.