Let H be a subgroup of the fundamental group π1(X,x0). In this paper, first, we give a positive answer to a problem presented by Brodskiy et al. in [8, Problem 4.8]. Second, by extending the concept of strong SLT space to a relative version with respect to H, strong H-SLT space, we investigate the existence of a covering map for strong H-SLT spaces. Moreover, we show that a semicovering map is a covering map in the presence of strong H-SLT property. Among other things, we present conditions under which the whisker topology agrees with the lasso topology on X˜H. Also, we study the relationship between open subsets of π1wh(X,x0) and π1l(X,x0). Finally, we give several examples to justify the definition and study of strong H-SLT spaces when H is a non-trivial subgroup.

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