Let H be a subgroup of π 1 (X, x 0 ). In this paper, we extend the concept of X being SLT space to H-SLT space at x 0 . First, we show that the fibers of the endpoint projection p H :X H → X are topological group when X is H-SLT space at x 0 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 p H has the unique path lifting property if and only if H is a closed normal subgroup of π qtop 1 (X, x 0 ) when X is SLT at x 0 . Second, we present conditions under which the whisker topology is agree with the quotient of compact-open topology onX H . Also, we study the relationship between open subsets of π wh 1 (X, x 0 ) and π qtop 1 (X, x 0 ).

In this paper, by introducing some kind of small loop transfer spaces at a point, we study the behavior of topologized fundamental groups with the compact-open topology and the whisker topology, π qtop 1 (X, x 0 ) and π wh 1 (X, x 0 ), respectively. In particular, we give necessary or sufficient conditions for coincidence and being topological group of these two topologized fundamental groups. Finally, we give some examples to show that the reverse of some of these implications do not hold, in general.

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It is important to classify covering subgroups of the fundamental group of a topological space using their topological properties in the topologized fundamental group. In this paper, we introduce and study some topologies on the fundamental group and use them to classify coverings, semicoverings, and generalized coverings of a topological space. To do this, we use the concept of subgroup topology on a group and discuss their properties. In particular, we explore which of these topologies make the fundamental group a topological group. Moreover, we provide some examples of topological spaces to compare topologies of fundamental groups.

By generalizing the whisker topology on the nth homotopy group of pointed space (X, x 0), denoted by π wh n (X, x 0), we show that π wh n (X, x 0) is a topological group if n ≥ 2. Also, we present some necessary and sufficient conditions for π wh n (X, x 0) to be discrete, Hausdorff and indiscrete. Then we prove that L n (X, x 0) the natural epimorphic image of the Hawaiian group H n (X, x 0) is equal to the set of all classes of convergent sequences to the identity in π wh n (X, x 0). As a consequence, we show that L n (X, x 0) ∼ = L n (Y, y 0) if π wh n (X, x 0) ∼ = π wh n (Y, y 0), but the converse does not hold in general, except for some conditions. Also, we show that on some classes of spaces such as semilocally n-simply connected spaces and n-Hawaiian like spaces, the whisker topology and the topology induced by the compact-open topology of n-loop space coincide. Finally, we show that n-SLT paths can transfer π wh n and hence L n isomorphically along its points.