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Consider the following two structure-adding refinements of the fundamental group of a topological space: the set $\pi_1(X)$ inherits a quotient topology from the compact-open topology of $X^{S^1}$, ...

It is well knows that for a nice (locally path connected, semi-locally simply connected) topological spaces, the category of covering spaces over $X$ is equivalent to the functor category $\left[\Pi_1\...