Some generalized metric properties of $n$-semitopological groups

TL;DR

This paper explores generalized metric properties of n-semitopological groups, revealing conditions under which they exhibit semi-metrizability, topological group structure, and analyzing their cardinal invariants.

Contribution

It introduces and studies properties of n-semitopological groups, extending known classes and discussing their metric and topological characteristics.

Findings

Hausdorff first-countable 2-semitopological groups are semi-metrizable

Locally compact, Baire, σ-compact 2-semitopological groups are topological groups

Cardinal invariants of n-semitopological groups are analyzed

Abstract

A semitopological group $G$ is called {\it an $n$-semitopological group}, if for any $g\in G$ with $e\not\in\overline{\{g\}}$ there is a neighborhood $W$ of $e$ such that $g\not\in W^{n}$, where $n\in\mathbb{N}$. The class of $n$-semitopological groups ($n\geq 2$) contains the class of paratopological groups and Hausdorff quasi-topological groups. Fix any $n\in\mathbb{N}$. Some properties of $n$-semitopological groups are studied, and some questions about $n$-semitopological groups are posed. Some generalized metric properties of $n$-semitopological groups are discussed, which contains mainly results are that (1) each Hausdorff first-countable 2-semitopological group admits a coarsersemi-metrizable topology; (2) each locally compact, Baire and $\sigma$-compact 2-semitopological group is a topological group; (3) the condensation of some kind of 2-semitopological groups topologies are given. Finally, some cardinal invariants of $n$-semitopological groups are discussed.