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 $S$.Comment: 4 page

On the length of chains of proper subgroups covering a topological group

We prove that if an ultrafilter L is not coherent to a Q-point, then each analytic non-sigma-bounded topological group G admits an increasing chain <G_a : a of its proper subgroups such that: (i) U_{a in b(L)} G_a=G; and $(ii)$ For every sigma-bounded subgroup H of G there exists a such that H is a subset of G_a. In case of the group Sym(w) of all permutations of w with the topology inherited from w^w this improves upon earlier results of S. Thomas

A homogeneous space whose complement is rigid

We construct a homogeneous subspace of 2ω whose complement is dense in 2ω and rigid. Using the same method, assuming Martin’s Axiom, we also construct a countable dense homogeneous subspace of 2ω whose complement is dense in 2ω and rigid.The first-listed and third-listed authors were supported by the FWF grant I 1209-N25.The second-listed author acknowledges generous hospitality and support from the Kurt Gödel Research Center for Mathematical Logic.The third-listed author also thanks the Austrian Academy of Sciences for its generous support through the APART Program