This paper is devoted to study some topological properties of the SG subgroup, π sg 1 (X, x), of the quasitopological fundamental group of a based space (X, x), π qtop 1 (X, x), its topological properties as a subgroup of the topological fundamental group π τ 1 (X, x) and its influence on the existence of universal covering of X. First, we introduce small generated spaces which have indiscrete topological fundamental groups and also small generated coverings which are universal coverings in the categorical sense. Second, we give a necessary and sufficient condition for the existence of the small generated coverings. Finally, by introducing the notion of semi-locally small generatedness we show that the quasitopological fundamental groups of semi-locally small generated spaces are topological groups.

In this paper we devote to spaces that are not homotopically hausdorff and study their covering spaces. We introduce the notion of small covering and prove that every small covering of X is the universal covering in categorical sense. Also, we introduce the notion of semi-locally small loop space which is the necessary and sufficient condition for existence of universal cover for non-homotopically hausdorff spaces, equivalently existence of small covering spaces. Also, we prove that for semi-locally small loop spaces, X is a small loop space if and only if every cover of X is trivial if and only if π top 1 (X) is an indiscrete topological group.2010 Mathematics Subject Classification. 57M10, 57M05, 55Q05, 57M12.

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H. Fischer et al. (Topology and its Application, 158 (2011) 397-408.) introduced the Spanier group of a based space (X, x) which is denoted by π sp 1 (X, x). By a Spanier space we mean a space X such that π sp 1 (X, x) = π 1 (X, x), for every x ∈ X. In this paper, first we give an example of Spanier spaces. Then we study the influence of the Spanier group on covering theory and introduce Spanier coverings which are universal coverings in the categorical sense. Second, we give a necessary and sufficient condition for the existence of Spanier coverings for non-homotopically path Hausdorff spaces. Finally, we study the topological properties of Spanier groups and find out a criteria for the Hausdorffness of topological fundamental groups.

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 ).