More on convergent sequences in free topological groups

We show that, for every infinite cardinal τ, there exists a zero-dimensional pseudocompact space X with | X | = τ ω such that all countable subsets of X are closed (hence all compact subsets of X are finite), but both the free Abelian topological group A ( X ) and the free topological group F ( X ) on X contain a compact subspace of cardinality τ. This answers a recent question raised by A.V. Arhangelʼskii. We also show that there exists a countably compact space Y without infinite compact subsets such that the groups F ( Y ) and A ( Y ) contain the one-point compactification of a discrete space of cardinality 2 c (under additional set-theoretic assumptions, the groups F ( Y ) and A ( Y ) can contain arbitrarily big compact subsets). wever, X is homeomorphic to a topological group, then the existence of an infinite compact subset of F ( X ) (or A ( X ) ) implies that X contains an infinite compact subset as well.