Some generalized metric properties of $n$-semitopological groups
Fucai Lin, Xixi Qi
TL;DR
The paper introduces and analyzes $n$-semitopological groups, a broad generalization of almost paratopological and quasi-topological groups, and investigates their generalized metric properties and cardinal invariants. It shows how $m$-semitopological structure can be characterized by the closedness of certain $m$-tuples with product equal to the identity, and uses quotient constructions to identify when such groups yield topological groups under additional hypotheses; it also demonstrates that Hausdorff $2$-semitopological groups can admit coarser semi-metrizable or submetrizable topologies and studies condensation and cellularity to derive cardinality bounds. The work discusses the behavior of several cardinal invariants and poses numerous open problems guiding future research in the metric theory of generalized groups.
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 $σ$-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.