Covering space

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 étale 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.^[1]: 10

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).^[2]: 29 Under certain conditions, covering spaces also exhibit a Galois correspondence with the subgroups of the fundamental group.