Small loop transfer spaces with respect to subgroups of fundamental groups

Let H be a subgroup of π1(X,x0). In this paper, we extend the concept of X being SLT space to H-SLT space at x0. First, we show that the fibers of the endpoint projection pH:X˜H→X are a topological group when X is H-SLT space at x0 and H is a normal subgroup. Also, we show that under these conditions the concepts of homotopically path Hausdorff relative to H and homotopically Hausdorff relative to H coincide. Moreover, among other things, we show that the endpoint projection map pH has the unique path lifting property if and only if H is a closed normal subgroup of π1qtop(X,x0) when X is H-SLT at x0. Second, we present conditions under which the whisker topology agrees with the quotient of compact-open topology on X˜H. Also, we study the relationship between open subsets of π1wh(X,x0) and π1qtop(X,x0).