On the Spanier Groups and Covering and Semicovering Spaces

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.