Metrizability of Clifford topological semigroups (Q444659)

By the Birkhoff-Kakutani theorem, a topological group \(G\) is metrizable if and only if it is first countable. In this paper, the authors prove that a countably compact Clifford topological semigroup \(S\) is metrizable if and only if the set \(E\) of idempotents of \(S\) is a metrizable \(G_\delta\)-set in \(S\). The proof is divided into a series of the following claims: \textbf{Claim 1:} The space \(S\) is first countable at each point \(e\in E\). \textbf{Claim 2:} The inversion \(x\mapsto x^{-1}:S\rightarrow S\) is continuous at each \(e\in E\). \textbf{Claim 3:} For every idempotent \(e\in E\) the maximal subgroup \(H_e=\{x\in S:xx^{-1}=e\}\) is a metrizable topological group. \textbf{Claim 4:} The semigroup \(S\) is topologically periodic, i.e., if for each element \(x\in S\) and each neighborhood \(O_x\subseteq S\) of \(x\) there is an integer \(n\geq 2\) with \(x^n\in O_x\). \textbf{Claim 5:} \(S\) is a topological Clifford semigroup.