Metrizability of Clifford topological semigroups

TL;DR

This paper establishes a precise criterion for when a Clifford topological semigroup is metrizable, linking it to the properties of being an M-space and having a metrizable set of idempotents, applicable also to countably compact cases.

Contribution

It provides a new metrization criterion for Clifford topological semigroups based on M-space and the metrizability of the idempotent set, extending to countably compact cases.

Findings

Metrizability characterized by M-space and idempotent set properties

Criterion applies to countably compact Clifford semigroups

Set of idempotents is a metrizable G_delta-set

Abstract

We prove that a topological Clifford semigroup $S$ is metrizable if and only if $S$ is an $M$-space and the set $E=\{e\in S:ee=e\}$ of idempotents of $S$ is a metrizable $G_\delta$-set in $S$. The same metrization criterion holds also for any countably compact Clifford topological semigroup $S$.