Let $H$ be a subgroup of the fundamental group $\pi_{ 1}(X,x_{ 0})$. By extending the concept of strong SLT space to a relative version with respect to $H$, strong $H$-SLT space, first, 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. Second, we present conditions under which the whisker topology agrees with the lasso topology on $\widetilde{ X}_{ H}$. Also, we study the relationship between open subsets of $\pi_{ 1}^{ wh}(X,x_{ 0})$ and $\pi_{ 1}^{ l}(X,x_{ 0})$. Finally, we give some examples to justify the definition and study of strong $H$-SLT spaces.
Categories: math.AT