We introduce and study oscillator topologies on paratopological groups and define certain related number invariants. As an application we prove that a Hausdorff paratopological group G admits a weaker Hausdorff group topology provided G is 3-oscillating. A paratopological group G is 3-oscillating (resp. 2-oscillating) provided for any neighborhood U of the unity e of G there is a neighborhood V...

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 precompa...