Products of topological groups and ω-stability

Abstract

We present conditions on factors guaranteeing that the product of two spaces (topological groups) is (fairly) pseudo-ℵ1-compact or ω-stable. It is shown, for example, that the product X×Y of a regular pseudo-ℵ1-compact P-space X with a weakly Lindelöf space Y is fairly pseudo-ℵ1-compact. Similarly, the product G×Y of a pseudo-ℵ1-compact P-group G and a fairly pseudo-ℵ1-compact space Y is fairly pseudo-ℵ1-compact.Also, we prove that for each infinite cardinal τ, a Tychonoff space X is τ-stable if and only if the free topological group F(X) (equivalently, the free Abelian topological group A(X)) on X is τ-steady. This enables us to establish τ-stability of several classes of Tychonoff spaces. Additionally, we resolve Open Problem 5.6.2 from [2] by proving that the product G×H of a Lindelöf Σ-group G and a Lindelöf P-group H is τ-stable provided τ=ℵn for some n∈ω or τω=τ.