A note on compact-like semitopological groups

Keywords:
semitopological group, paratopological group, compact-like semitopological group, compact-like paratopological group, continuity of the inverse, joint continuity, separation axioms, countably compact paratopological group, feebly compact topological groupAbstract
We present a few results related to separation axioms and automatic continuity of operations in compact-like semitopological groups. In particular, is provided a semiregular semitopological group which is not . We show that each weakly semiregular compact semitopological group is a topological group. On the other hand, constructed examples of quasiregular compact and sequentially compact quasitopological groups, which are not paratopological groups. Also we prove that a semitopological group is a topological group provided there exists a Hausdorff topology on such that is a precompact topological group and is weakly semiregular or is a feebly compact paratopological group and is .