Properties of locally semi-compact Ir-topological groups (Q6611491)

A subset \(A\) of topological space \(X\) is semi-open if there is a subset \(B\) of \(X\) such that \(B \subseteq A \subset \overline{B},\) where \(\overline{B}\) is the closure of \(B\) in \(X.\) A mapping \(f : X \to Y\) between two topological spaces is irresolute if for any semi-open subset \(V\) of \(Y\) its preimage \(f^{-1}(V)\) is semi-open in \(X.\) Analogous to topological groups, one can consider the so-called Ir-topological groups, that is groups with topology whose operations of multiplication and taking of inverses are irresolute mappings.\N\NIn the paper under review, the authors establish some properties of locally semi-compact \(Ir\)-topological groups. Clearly, local semi-compactness generalizes local compactness: a topological space \(X\) is semi-compact if every cover of \(X\) by semi-open sets has a finite subcover and \(X\) is locally semi-compact if every its point has an open semi-compact neighbourhood.\N\NThe main results of the considered paper include the following:\N\NTheorem 3.10. Suppose \(G\) is an Ir-topological group and each semi-open set in \(G\) is pre-open, i.e. is contained in the interior of its closure. If \(A\) is a locally semi-compact pre-open subgroup, then \(A\) is a regular-open (meaning that \(A\) coincides with the interior of its closure) Ir-topological subgroup.\N\NTheorem 3.11. Suppose \(G\) is a locally semi-compact Ir-topological group and each semi-open set is pre-open. If \(Y\) is an open subgroup of \(G\) then \(Y\) is a locally semi-compact Ir-topological group.\N\NTheorem 3.15. Suppose \(G\) is a locally semi-compact Ir-topological group and each semi-open set in \(G\) is pre-open. If \(A\) is an invariant subgroup, then \((G/A, \mathcal{A})\) is a locally semi-compact Ir-topological group.