# Topologized standard construction and locally quasinormal subgroups

## TL;DR

This paper investigates the topological properties of subgroups of the fundamental group, focusing on conditions for homotopically Hausdorff properties, topology coincidences, and the broader class of locally quasinormal subgroups.

## Contribution

It introduces weaker conditions for homotopically Hausdorff properties relative to subgroups and explores the topology of standard constructions for locally quasinormal subgroups.

## Key findings

- Conditions under which homotopically Hausdorff implies homotopically path Hausdorff
- Coincidence of whisker and quotient topologies on fundamental groups
- Locally quasinormal subgroups are more extensive than normal subgroups

## Abstract

This paper is the extended version of some results in [13, 14]. Let H be a subgroup of fundamental group. The first paper of the paper is devoted to studying weaker conditions under which homotopically Hausdorff relative to H becomes homotopically path Hausdorff relative to H. By using of these conditions, we explore the connection between whisker and quotient topologies on fundamental group. After that, we address the coincidence of two determined topologies on the standard construction XeH when H is a locally quasinormal subgroup. Finally Example 3.14 illustrates that these kinds of subgroups are more extensive than normal subgroups and justifies the generalizations of these results.