Mikhail Tkachenko

When is an open subset of a Lindelöf topological group homeomorphic to a (para)topological group?‎‎

In 2023, A. V. Arhangel’skii proved the following:

Theorem. If G is a Lindelöf topological group with countable cellularity and F\subset G is a closed subset such that the complement O=G\setminus F is homeomorphic to a topological group, then F is a zero-set in G.

We will present a short and elegant proof of this result based on the original Arhangel’skii’s arguments.

The main issue to be discussed is whether Arhangel’skii’s theorem can be extended to the more general situation in which the complement G\setminus F is homeomorphic to a paratopological group. Apparently, this does not sound very much different from the purely “topological group” case considered in the above theorem. We will attempt to dispel this illusion, demonstrate that such an extension can be done if G is a \sigma-compact (or, more generally, Lindelöf \Sigma-) topological group, and describe the techniques used in our arguments.