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