Band of topological groups

TL;DR

This paper explores the structure of bands of topological groups derived from cryptogroups, establishing conditions for metrizability and Hausdorff properties related to normal subcryptogroups.

Contribution

It introduces the concept of a band of topological groups from cryptogroups and characterizes their metrizability and quotient space properties.

Findings

A band of topological groups is metrizable iff each $\

6$-class is metrizable.

The quotient $S/N$ is Hausdorff iff $ ho_{_N}$ is closed in $S imes S$.

Abstract

In this article, we construct a band of topological groups from a cryptogroup. Also, we prove that a band of topological groups is metrizable if and only if each $\mathcal{H}$-class is metrizable. Finally, we demonstrate that if $S$ is a band of topological groups and $N$ is a full normal subcryptogroup of $S$, then $S/N$ is Hausdorff if and only if $\rho_{_N}$ is closed in $S \times S$ if and, $\rho_{_N}$ is closed in $S \times S$ if and only if $N$ is closed in $S$.