# A note on compact-like semitopological groups

## TL;DR

This paper explores separation axioms and automatic continuity in compact-like semitopological groups, providing examples and conditions under which such groups are topological or quasitopological.

## Contribution

It presents new examples of semitopological groups with specific separation properties and establishes conditions for when these groups are topological.

## Key findings

- A semiregular semitopological group can be non-$T_3$.
- Weakly semiregular compact semitopological groups are topological.
- Examples of quasiregular $T_1$ and $T_2$ compact quasitopological groups that are not paratopological.

## 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 $T_3$.