> [!definition] > > Let $X$ be a [[Topological Space|topological space]]. A **covering space** of $X$ is a topological space $\td X$ and a [[Continuity|continuous]] map $p: \td X \to X$ such that for each $x \in X$, there exists an [[Neighbourhood|open neighbourhood]] $U \in \cn^o(x)$ where $p^{-1}(U)$ is a disjoint union of [[Open Set|open]] sets in $\td X$, each of which is mapped homeomorphically onto $U$ by $p$.