Local homeomorphism

In mathematics, more specifically topology, a local homeomorphism is intuitively a function, f, between topological spaces that preserves local structure.

Formal definition

Let X and Y be topological spaces. A function is a local homeomorphism if for every point x in X there exists an open set U containing x, such that the image is open in Y and the restriction is a homeomorphism.

Examples

By definition, every homeomorphism is also a local homeomorphism.

If U is an open subset of Y equipped with the subspace topology, then the inclusion map i : U → Y is a local homeomorphism. Openness is essential here: the inclusion map of a non-open subset of Y never yields a local homeomorphism.

Every covering map is a local homeomorphism; in particular, the universal cover p : C → Y of a space Y is a local homeomorphism. In certain situations the converse is true. For example : if X is Haudorff and Y is locally compact and Hausdorff and p : X → Y is a proper local homeomorphism, then p is a covering map.