A note on compact-like semitopological groups

Abstract

The note contains a few results related to separation axioms and automatic continuity of operations in compact-like semitopological groups. In particular, is presented a semiregular semitopological group $G$ which is not $T_3$. We show that each weakly semiregular compact semitopological group is a topological group. On the other hand, constructed examples of quasiregular $T_1$ compact and $T_2$ sequentially compact quasitopological groups, which are not paratopological groups. Also we prove that a semitopological group $(G,\tau )$ is a topological group provided there exists a Hausdorff topology $\sigma \supset \tau$ on $G$ such that $(G,\sigma )$ is a precompact topological group and $(G,\tau )$ is weakly semiregular or $(G,\sigma )$ is a feebly compact paratopological group and $(G,\tau )$ is $T_3$.

9 pages; minor updates and corrections. arXiv admin note: text overlap with arXiv:1003.5343