Examples of quasitopological groups (Q2926306)

A group \(G\) with a Tychonoff topology is said to be quasitopological if the multiplication is separately continuous and the inverse operation is continuous. The authors give several examples of quasitopological groups with topology containing a metrizable topology that are not topological groups. They consider three topologies on the additive group \(\mathbb R\times\mathbb R\) (finer than the Euclidean topology) giving quasitopological groups and they examine some of the subgroups. For a metrizable \(0\)-dimensional commutative topological group \(E\) without isolated points in which there is a nontrivial sequence converging to the neutral element they are able to define quasitopological subgroups of \(E\times E\). The constructed examples in particular include a countable not first countable quasitopological group \(G\) with countable \(\pi\)-weight, countable tightness, and countable \(\delta\)-character and a countable quasitopological group \(P\) with countable \(\pi\)-weight, countable tightness, and an uncountable \(\delta\)-character.