Covering space - Knowlpedia

Let $X$ and $\widetilde X$ be topological spaces. A covering space of $X$ is a topological space $\widetilde X$ , a continuous surjection

p:\widetilde X\to X,

and the following local condition: for every $x\in X$ , there is an open neighborhood $U\subseteq X$ such that

p^{-1}(U)=\coprod_{\alpha\in A}V_\alpha

is a disjoint union of open subsets $V_\alpha\subseteq\widetilde X$ , and for every $\alpha$ the restriction

p|_{V_\alpha}:V_\alpha\xrightarrow{\cong}U

is a homeomorphism. Such a $U$ is evenly covered, and the $V_\alpha$ are its sheets. The map $p$ is the covering map.

Fibers and sheets

Every fiber $p^{-1}(x)$ is discrete in the subspace topology. A covering is $n$ -sheeted if every fiber has $n$ points. A one-sheeted covering is a homeomorphism, while a disconnected covering may have several connected components lying over the same base.

Maps and examples

A map of covering spaces over $X$ is a continuous map $F:\widetilde X\to\widetilde Y$ satisfying $p_Y\circ F=p_X$ . The product projection $X\times F\to X$ , with $F$ nonempty and discrete, is a covering space, and the projection $\mathbb R\to S^1$ , $t\mapsto e^{2\pi i t}$ , is an infinite-sheeted covering.

The evenly covered condition is stronger than being a local homeomorphism: it requires the full inverse image $p^{-1}(U)$ to split into disjoint sheets, each mapping homeomorphically onto the same $U$ .

References

Allen Hatcher, Algebraic Topology, Cambridge University Press, 2002, Chapter 1, §1.3, “Covering Spaces,” pp. 56–83. Author-hosted PDF.