For a connected, locally path connected space X , let H be a subgroup of the fundamental group of X , p1(X ; x ) . We show that there exists an open cover U of X such that H contains the Spanier group p (U ; x ) if and only if the core of H in p1(X ; x ) is open in the quasitopological fundamental group p qt o p 1 ( X ; x ) or equivalently it is open in the topological fundamental group p t 1 ( X ; x ). As a con-sequence, 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 p qt o p 1 (X ; x ) .

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