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 ${\displaystyle f:X\to Y}$ is a local homeomorphism, ${\displaystyle X}$ is said to be an étale space over ${\displaystyle Y.}$ Local homeomorphisms are used in the study of sheaves. Typical examples of local homeomorphisms are covering maps.

A topological space ${\displaystyle X}$ is locally homeomorphic to ${\displaystyle Y}$ if every point of ${\displaystyle X}$ has a neighborhood that is homeomorphic to an open subset of ${\displaystyle Y.}$ For example, a manifold of dimension ${\displaystyle n}$ is locally homeomorphic to ${\displaystyle \mathbb {R} ^{n}.}$

If there is a local homeomorphism from ${\displaystyle X}$ to ${\displaystyle Y,}$ then ${\displaystyle X}$ is locally homeomorphic to ${\displaystyle Y,}$ but the converse is not always true. For example, the two dimensional sphere, being a manifold, is locally homeomorphic to the plane ${\displaystyle \mathbb {R} ^{2},}$ but there is no local homeomorphism ${\displaystyle S^{2}\to \mathbb {R} ^{2}.}$