Universal Cover

Covering Map

A covering map (also called a projection) is a surjective continuous map $p:\tilde{X}\to X$ that each point $x\in X$ has a neighborhood $U$ such that

• $p^{-1}(U)$ is a disjoint union of open sets in $\tilde{X}$: $p^{-1}(U)={⨆}_{\alpha \in A}{V}_{\alpha }$,
• for each $\alpha \in A$, the restriction $p{|}_{V_{\alpha }}:V_{\alpha }\to U$ is a homeomorphism.

Such a neighborhood $U$ is called evenly covered by $p$.

e.g. The exponential map $p:\mathbb{R}\to S^1$ defined by $p(t)=e^{2\pi it}$ is a covering map, wrapping the real line around the circle infinitely many times. Locally, it looks like $\mathbb{R}\to S^1$ is a bunch of evenly spaced intervals mapping homeomorphically onto an arc of the circle. A concrete application of this can be seen in the proof of the fundamental group of $S^1$.

Universal Cover

A universal cover of a topological space $X$ is a covering space $\tilde{X}$ with a covering map $p:\tilde{X}\to X$ such that $\tilde{X}$ is simply connected.