Quotient spaces with strong subgyrogroups

TL;DR

This paper studies the properties of quotient spaces formed from strongly topological gyrogroups and their subgyrogroups, establishing conditions under which various topological properties are preserved or equivalent.

Contribution

It provides new characterizations of quotient spaces in strongly topological gyrogroups and links properties of the quotient to the properties of the original gyrogroup.

Findings

G/H is first-countable iff it is bisequential, weakly first-countable, and csf-countable sequential.

Sequentiality of G is implied by sequential G/H when H is locally compact and metrizable.

G/H being cosmic implies G is cosmic if H is closed, first-countable, and separable.

Abstract

In this paper, we mainly investigate the quotient spaces G/H when G is a strongly topological gyrogroup and H is a strong subgyrogroup of G. It is shown that if G is a strongly topological gyrogroup, H is a closed strong subgyrogroup of G and H is inner neutral, then the quotient space G/H is first-countable if and only if G/H is a bisequential space if and only if G/H is a weakly first-countable space if and only if G/H is a csf-countable and sequential a7-space. Moreover, it is shown that if H is a locally compact metrizable strong subgyrogroup of G and the quotient space G/H is sequential, then G is also sequential; if H is a closed first-countable and separable strong subgyrogroup of G, the quotient space G/H is a cosmic space, then G is also a cosmic space; if the quotient space G/H has a star-countable cs-network or star-countable wcs*-network, then G also has a star-countable cs-network or star-countable wcs*-network, respectively.