Local Homeomorphism

In mathematics, more specifically topology, a local homeomorphism is intuitively a function, f, between topological spaces that preserves local structure. Equivalently, one can cover the domain of this function by open sets, such that f restricted to each such open set is a homeomorphism onto its image. In particular, every homeomorphism is a local homeomorphism. The formal definition is given below.

Local homeomorphisms are very important in mathematics, particularly in the theory of manifolds (e.g. differential topology) and algebraic topology. An important example of local homeomorphisms are covering maps. Covering maps are important, because they satisfy the local triviality condition, inducing isomorphisms of particular homotopy groups, and in particular, one can lift any path in the base space of a covering map, to a path in the total space. Although local homeomorphisms are not as strong as covering maps in this respect, they have many important applications in differential topology; namely one uses the notion of a local homeomorphism to define a differentiable manifold.