Topologized standard construction and locally quasinormal subgroups

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.