Feebly compact paratopological groups and real-valued functions

Abstract

We present several examples of feebly compact Hausdorff paratopological groups (i.e., groups with continuous multiplication) which provide answers to a number of questions posed in the literature. It turns out that a 2-pseudocompact, feebly compact Hausdorff paratopological group $$G$$ can fail to be a topological group. Our group $$G$$ has the Baire property, is Fréchet–Urysohn, but it is not precompact. It is well known that every infinite pseudocompact topological group contains a countable non-closed subset. We construct an infinite feebly compact Hausdorff paratopological group $$G$$ all countable subsets of which are closed. Another peculiarity of the group $$G$$ is that it contains a nonempty open subsemigroup $$C$$ such that $$C^{-1}$$ is closed and discrete, i.e., the inversion in $$G$$ is extremely discontinuous. We also prove that for every continuous real-valued function $$g$$ on a feebly compact paratopological group $$G$$ , one can find a continuous homomorphism $$\varphi $$ of $$G$$ onto a second countable Hausdorff topological group $$H$$ and a continuous real-valued function $$h$$ on $$H$$ such that $$g=h\circ \varphi $$ . In particular, every feebly compact paratopological group is $$\mathbb{R }_3$$ -factorizable. This generalizes a theorem of Comfort and Ross established in 1966 for real-valued functions on pseudocompact topological groups.