Metrizability of Clifford topological semigroups

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 $G_δ$-set in $S$. The same metrization criterion holds also for any countably compact Clifford topological semigroup $S$.

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