Some properties involving feeble compactness. I: Compact-boundedness and selective feeble compactness (Q6551732)

The authors define some subclasses with property \(\mathcal{P}_0\) of feebly compact Hausdorff spaces such as the (weakly) compact-bounded spaces and selectively (sequentially) feebly compact spaces and collect some basic results of spaces with \(\mathcal{P}_0\) and results that establish the relationship between spaces with \(\mathcal{P}_0\). Then they focus on finding conditions under which a paratopological group with \(\mathcal{P}_0\) is a topological group. They also study how to extend the properties of the quotient space \(G/H\) to \(G\) where \(H\) is either a compact, locally compact or a (weakly) compact-bounded subgroup of a topological group \(G\). Finally, they show that feeble compactness and sequential pseudocompactness are equivalent properties in the class of paratopological Hausdorff groups.