ar X iv : 0 81 0 . 30 28 v 1 [ m at h . G N ] 1 6 O ct 2 00 8 OSCILLATOR TOPOLOGIES ON A PARATOPOLOGICAL GROUP AND RELATED NUMBER INVARIANTS

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 ⊂ G of e such that V −1 V V −1 ⊂ U U −1 U (resp. V −1 V ⊂ U U −1). The class of 2-oscillating paratopolog-ical groups includes all collapsing, all nilpotent paratopological groups, all paratopo-logical groups satisfying a positive law, all paratopological SIN-group and all saturated paratopological groups (the latter means that for any nonempty open set U ⊂ G the set U −1 has nonempty interior). We prove that each totally bounded paratopological group G is countably cellular; moreover, every cardinal of uncountable cofinality is a precal-iber of G. Also we give an example of a saturated paratopological group which is not isomorphic to its mirror paratopological group as well as an example of a 2-oscillating paratopological group whose mirror paratopological group is not 2-oscillating.