  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 $(X,x)$, $\pt$, 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.
