This paper is devoted to study some topological properties of the SG subgroup, πsg1 (X, x), of the quasitopological fundamental group of a based space (X, x), πqtop1 (X, x), its topological properties as a subgroup of the topological fundamental group πτ1 (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.

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