# covering space

Let $X$ and $E$ be topological spaces  and suppose there is a surjective continuous map $p\colon E\rightarrow X$ which satisfies the following condition: for each $x\in X$, there is an open neighborhood $U$ of $x$ such that

• $p^{-1}(U)$ is a disjoint union of open sets $E_{i}\subset E$, and

• each $E_{i}$ is mapped homeomorphically onto $U$ via $p$.

Then $E$ is called a covering space, $p$ is called a covering map, the $E_{i}$’s are sheets of the covering of $U$ and for each $x\in X$, $p^{-1}(x)$ is the fiber of $p$ above $x$. The open set $U$ is said to be evenly covered. If $E$ is simply connected, it is called the universal covering space.

From this we can derive that $p$ is a local homeomorphism, so that any local property $E$ has is inherited by $X$ (local connectedness, local path connectedness etc.). Covering spaces are foundational in the study of the fundamental group  of a topological space; in particular, there is a correspondence (http://planetmath.org/ClassificationOfCoveringSpaces) between connected coverings of $X$ and subgroups of the group of deck transformations  of its universal covering space which is exactly analogous to the fundamental theorem of Galois theory  .

Covering maps are especially important in the study of Riemann surfaces; in this context, one sometimes discusses a generalized notion of covering map called a “ramified covering”; this allows one to replace a discrete set of the local homeomorphisms by maps that locally look like $z\mapsto z^{n}$ in the complex plane  near zero. Covering maps are also generalized in algebraic geometry   ; there the corresponding notion is that of étale morphism.

Note that this is a completely separate usage of the word “cover” than we encounter in “open cover”; confusion usually does not arise.

Title covering space CoveringSpace 2013-03-22 11:59:30 2013-03-22 11:59:30 rmilson (146) rmilson (146) 21 rmilson (146) Definition msc 55R05 covering map Cover Site EtaleMorphism