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 f : X → Y is a local homeomorphism, X is said an étale space over Y. Local homeomorphisms are used in study of sheaves. In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure. If f : X → Y is a local homeomorphism, X is said an étale space over Y. Local homeomorphisms are used in study of sheaves. A topological space X is locally isomorphic to Y if every point of X has a neighborhood that is homeomorphic to an open subset of Y. For example, a manifold of dimension n is locally isomorphic to R n . {displaystyle mathbb {R} ^{n}.} If there is a local homeomorphism from X to Y, then X is locally isomorphic to Y, but the converse is not always true. For example, the two dimensional sphere, being a manifold, is locally homeomorphic to the plane R 2 , {displaystyle mathbb {R} ^{2},} but there is no local homeomorphism between them (in either direction). Let X and Y be topological spaces. A function f : X → Y is a local homeomorphism if for every point x in X there exists an open set U containing x, such that the image f(U) is open in Y and the restriction f|U : U → f(U) is a homeomorphism (where the respective subspace topologies are used on U and on f(U)).