Feebly compact paratopological groups and real-valued functions

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 φ of G onto a second countable Hausdorff topological group H and a continuous real-valued function h on H such that g = h ◦φ . In particular, every feebly compact paratopological group is R3-factorizable. This generalizes a theorem of Comfort and Ross established in 1966 for real-valued functions on pseudocompact topological groups.