Let \(X\) be a topological space. A universal cover of \(X\) is a covering space \(p\colon \tilde X\to X\) with \(\tilde X\) simply connected [@Hat02].
Let \(X\) be path-connected and locally path-connected. Then \(X\) has a universal cover if and only if \(X\) is semilocally simply connected [@Hat02], and a universal cover is unique up to isomorphism of covering spaces [@Hat02]. A universal cover covers every path-connected covering space of \(X\) [@Hat02], and its deck transformation group is isomorphic to \(\pi_1(X)\) [@Hat02].