In this paper, the set C(X) of all covering spaces of arcwise connected, locally arcwise and semilocally simply connected pointed topological spaces (X, x0) in investigated algebraically through the study of the fundamental groups ∏1(U, x0)) and sail/lattice and hence idempotenl semigroup structures on C(X) are obtained. The situation where ∏1(X, x0) is abelian and (X, x0) has a universal covering space is also studied and a lattice structure on C(X) is obtained. Moreover, the topological problem of classifying the isomorphism classes of covering space of (X, x0) is converted to an algebraic one by the determination of all subgroups of ∏1(X, x0). For the unit circle S1, C(S1) is examined and the lattice C* (S1), the set of oll nontrivial covering spaces of S1, is shown to be order isomorphic to the divisibility lattice of the positive inlegu. Finally, In the case when ∏1(X, x0) is Hamittonian. C(X) is also studied.