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

Selective versions of chain condition type properties

We study selective and game-theoretic versions of properties like the ccc, weak Lindelöfness and separability, giving various characterizations of them and exploring connections between these properties and some classical cardinal invariants of the continuum

Some new directions in infinite-combinatorial topology

We give a light introduction to selection principles in topology, a young subfield of infinite-combinatorial topology. Emphasis is put on the modern approach to the problems it deals with. Recent results are described, and open problems are stated. Some results which do not appear elsewhere are also included, with proofs.Comment: Small update

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

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