In topology, a covering or covering projection is a map between topological spaces that, intuitively, locally acts like a projection of multiple copies of a space onto itself. In particular, coverings are special types of local homeomorphisms. If ${\displaystyle p:{\tilde {X}}\to X}$ is a covering, ${\displaystyle ({\tilde {X}},p)}$ is said to be a covering space or cover of ${\displaystyle X}$, and ${\displaystyle X}$ is said to be the base of the covering, or simply the base. By abuse of terminology, ${\displaystyle {\tilde {X}}}$ and ${\displaystyle p}$ may sometimes be called covering spaces as well. Since coverings are local homeomorphisms, a covering space is a special kind of étalé space.

Covering spaces first arose in the context of complex analysis (specifically, the technique of analytic continuation), where they were introduced by Riemann as domains on which naturally multivalued complex functions become single-valued. These spaces are now called Riemann surfaces.

Covering spaces are an important tool in several areas of mathematics. In modern geometry, covering spaces (or branched coverings, which have slightly weaker conditions) are used in the construction of manifolds, orbifolds, and the morphisms between them. In algebraic topology, covering spaces are closely related to the fundamental group: for one, since all coverings have the homotopy lifting property, covering spaces are an important tool in the calculation of homotopy groups. A standard example in this vein is the calculation of the fundamental group of the circle by means of the covering of ${\displaystyle S^{1}}$ by ${\displaystyle \mathbb {R} }$ (see below). Under certain conditions, covering spaces also exhibit a Galois correspondence with the subgroups of the fundamental group.

Let ${\displaystyle X}$ be a topological space. A covering of ${\displaystyle X}$ is a continuous map

such that for every ${\displaystyle x\in X}$ there exists an open neighborhood ${\displaystyle U_{x}}$ of ${\displaystyle x}$ and a discrete space ${\displaystyle D_{x}}$ such that ${\displaystyle \pi ^{-1}(U_{x})}$ is the disjoint union ${\displaystyle \displaystyle \bigsqcup _{d\in D_{x}}V_{d}}$ and ${\displaystyle \pi |_{V_{d}}:V_{d}\rightarrow U_{x}}$ is a homeomorphism for every ${\displaystyle d\in D_{x}}$. The open sets ${\displaystyle V_{d}}$ are called sheets, which are uniquely determined up to homeomorphism if ${\displaystyle U_{x}}$ is connected. For each ${\displaystyle x\in X}$ the discrete set ${\displaystyle \pi ^{-1}(x)}$ is called the fiber of ${\displaystyle x}$. If ${\displaystyle X}$ is connected (and ${\displaystyle {\tilde {X}}}$ is non-empty), it can be shown that ${\displaystyle \pi }$ is surjective, and the cardinality of ${\displaystyle D_{x}}$ is the same for all ${\displaystyle x\in X}$; this value is called the degree of the covering. If ${\displaystyle {\tilde {X}}}$ is path-connected, then the covering ${\displaystyle \pi :{\tilde {X}}\rightarrow X}$ is called a path-connected covering. This definition is equivalent to the statement that ${\displaystyle \pi }$ is a locally trivial fiber bundle.

Some authors also require that ${\displaystyle \pi }$ be surjective in the case that ${\displaystyle X}$ is not connected.

Since a covering ${\displaystyle \pi :E\rightarrow X}$ maps each of the disjoint open sets of ${\displaystyle \pi ^{-1}(U)}$ homeomorphically onto ${\displaystyle U}$ it is a local homeomorphism, i.e. ${\displaystyle \pi }$ is a continuous map and for every ${\displaystyle e\in E}$ there exists an open neighborhood ${\displaystyle V\subset E}$ of ${\displaystyle e}$, such that ${\displaystyle \pi |_{V}:V\rightarrow \pi (V)}$ is a homeomorphism.

It follows that the covering space ${\displaystyle E}$ and the base space ${\displaystyle X}$ locally share the same properties.