The strong Pytkeev property and strong countable completeness in (strongly) topological gyrogroups

TL;DR

This paper investigates the properties of topological gyrogroups, proving that sequential gyrogroups with an ^{\u00b5} e-base have the strong Pytkeev property and exploring conditions for Baire gyrogroups, also analyzing strongly countably complete gyrogroups.

Contribution

It establishes the strong Pytkeev property for sequential gyrogroups with an ^{\u00b5} e-base and characterizes strongly countably complete gyrogroups via subgyrogroups and quotient spaces.

Findings

Sequential gyrogroups with ^{e} e-base have the strong Pytkeev property.

Baire topological gyrogroups have equivalent conditions related to ^{e} e-base and strong Pytkeev property.

Strongly countably complete gyrogroups contain a closed, countably compact subgyrogroup with a metrizable quotient.

Abstract

A topological gyrogroup is a gyrogroup endowed with a topology such that the binary operation is jointly continuous and the inverse mapping is also continuous. In this paper, it is proved that if $G$ is a sequential topological gyrogroup with an $\omega^{\omega}$-base, then $G$ has the strong Pytkeev property. Moreover, some equivalent conditions about $\omega^{\omega}$-base and strong Pytkeev property are given in Baire topological gyrogroups. Finally, it is shown that if $G$ is a strongly countably complete strongly topological gyrogroup, then $G$ contains a closed, countably compact, admissible subgyrogroup $P$ such that the quotient space $G/P$ is metrizable and the canonical homomorphism $\pi :G\rightarrow G/P$ is closed.