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.

中文翻译：

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关于拓扑基础组和拓扑组的覆盖组

我们表明，每个拓扑组在标识元素上都是一个强大的小循环传递空间。这意味着对于一个连接的局部路径连接的拓扑组G，配备有由P（G，  e）上的紧凑开放拓扑诱导的商拓扑的通用路径空间\（\ widetilde {G} _ {e} \）为拓扑组。此外，我们证明有连接覆盖组的等价类之间的一对一对应ģ和亚组\（\ PI _ {1}（G，E）\）包含\（I _ *（ \ pi _ {1}（U，e））\）用于标识元素e的某个开放邻域U。