On Topologized Fundamental Groups and Covering Groups of Topological Groups

Abstract

We show that every topological group is a strong small loop transfer space at the identity element. This implies that for a connected locally path connected topological group G, the universal path space $$\widetilde{G}_{e}$$ equipped with the quotient topology induced by the compact-open topology on P(G, e) is a topological group. Moreover, we prove that there is a one-to-one correspondence between the equivalence classes of connected covering groups of G and the subgroups of $$\pi _{1}(G,e)$$ that contain $$i_*(\pi _{1}(U,e))$$ for some open neighborhood U of the identity element e.