  For a connected, locally path connected space $X$, let $H$ be a subgroup of
the fundamental group of $X$, $\pi_1(X,x)$. We show that there exists an open
cover $\cal U$ of $X$ such that $H$ contains the Spanier group $\pi({\U},x)$ if
and only if the core of $H$ in $\pi_1(X,x)$ is open in the quasitopological
fundamental group $\pi_1^{qtop}(X,x)$ or equivalently it is open in the
topological fundamental group $\pi_1^{\tau}(X,x)$. As a consequence, using the
relation between the Spanier groups and covering spaces, we give a
classification for connected covering spaces of $X$ based on the conjugacy
classes of subgroups with open core in $\pi_1^{qtop}(X,x)$. Finally, we give a
necessary and sufficient condition for the existence of a semicovering.
Moreover, we present a condition under which every semicovering of $X$ is a
covering.
