Local homeomorphism

In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure. If is a local homeomorphism, is said to be an étale space over Local homeomorphisms are used in the study of sheaves. Typical examples of local homeomorphisms are covering maps. A topological space is locally homeomorphic to if every point of has a neighborhood that is homeomorphic to an open subset of For example, a manifold of dimension is locally homeomorphic to If there is a local homeomorphism from to then is locally homeomorphic to but the converse is not always true. For example, the two dimensional sphere, being a manifold, is locally homeomorphic to the plane but there is no local homeomorphism A function between two topological spaces is called a if for every point there exists an open set containing such that the is open in and the restriction is a homeomorphism (where the respective subspace topologies are used on and on ). Local homeomorphisms versus homeomorphisms Every homeomorphism is a local homeomorphism. But a local homeomorphism is a homeomorphism if and only if it is bijective. A local homeomorphism need not be a homeomorphism. For example, the function defined by (so that geometrically, this map wraps the real line around the circle) is a local homeomorphism but not a homeomorphism. The map defined by which wraps the circle around itself times (that is, has winding number ), is a local homeomorphism for all non-zero but it is a homeomorphism only when it is bijective (that is, only when or ). Generalizing the previous two examples, every covering map is a local homeomorphism; in particular, the universal cover of a space is a local homeomorphism. In certain situations the converse is true. For example: if is a proper local homeomorphism between two Hausdorff spaces and if is also locally compact, then is a covering map.