Topological fundamental groups and small generated coverings

This paper is devoted to study some topological properties of the SG subgroup, ${\pi }_1^{sg}(X,x)$, of the quasitopological fundamental group of a based space $(X,x)$, $\text{pt}$, its topological properties as a subgroup of the topological fundamental group ${\pi }_1^{\tau }(X,x)$ 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.