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

Abstract

Atopological 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 !!-base, then G has the strong Pytkeev property. Moreover, some equivalent conditions about !!-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 ? : G ? G=P is closed.

A supplement on feathered gyrogroups

“…In [40,Theorem 3.5], the authors showed that if G is a sequential topological gyrogroup with an ω ω -base, then G has the strong Pytkeev property. Therefore, Corollary 3.3 poses the following result directly.…”

“…Clearly, every topological group is a topological gyrogroup and each topological gyrogroup is a rectifiable space. The readers may consult [5,6,10,12,13,14,15,16,39,40] for more details about topological gyrogroups.…”