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 $G_\delta$-set in $S$. The same metrization criterion holds also for any countably compact Clifford topological semigroup $S$. 
Taras Banakh, Oleg Gutik, Oles Potiatynyk, Alex Ravsky
math.GN math.GR