Editing Local homeomorphism (section)

{{Short description|Mathematical function revertible near each point}}
In [[mathematics]], more specifically [[topology]], a '''local homeomorphism''' is a [[Function (mathematics)|function]] between [[topological space]]s that, intuitively, preserves local (though not necessarily global) structure. 
If <math>f : X \to Y</math> is a local homeomorphism, <math>X</math> is said to be an '''étale space''' over <math>Y.</math>  Local homeomorphisms are used in the study of [[Sheaf (mathematics)|sheaves]]. Typical examples of local homeomorphisms are [[covering map]]s.

A topological space <math>X</math> is '''locally homeomorphic''' to <math>Y</math> if every point of <math>X</math> has a neighborhood that is [[homeomorphic]] to an open subset of <math>Y.</math> 
For example, a [[manifold]] of dimension <math>n</math> is locally homeomorphic to <math>\R^n.</math>

If there is a local homeomorphism from <math>X</math> to <math>Y,</math> then <math>X</math> is locally homeomorphic to <math>Y,</math> but the converse is not always true. 
For example, the two dimensional [[sphere]], being a manifold, is locally homeomorphic to the plane <math>\R^2,</math> but there is no local homeomorphism <math>S^2 \to \R^2.</math>