Definition 1 A covering space of a topological space \(X\) is a topological space \(C\) together with a continuous surjective map \(p: C \to X\) such that for every point \(x \in X\), there exists an open neighborhood \(U\) of \(x\) such that \(p^{-1}(U) \cong F \times U\) for some discrete set \(F\) (called the fiber over \(x\)), and the map \(p\) restricted to each component of \(p^{-1}(U)\) is a homeomorphism onto \(U\).
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