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 ${\displaystyle S^{2}\to \mathbb {R} ^{2}.}$

A function ${\displaystyle f:X\to Y}$ between two topological spaces is called a local homeomorphism if every point ${\displaystyle x\in X}$ has an open neighborhood ${\displaystyle U}$ whose image ${\displaystyle f(U)}$ is open in ${\displaystyle Y}$ and the restriction ${\displaystyle f{\big \vert }_{U}:U\to f(U)}$ is a homeomorphism (where the respective subspace topologies are used on ${\displaystyle U}$ and on ${\displaystyle f(U)}$).

Covering maps

Every homeomorphism is a local homeomorphism. The function ${\displaystyle \mathbb {R} \to S^{1}}$ defined by ${\displaystyle t\mapsto e^{it}}$ (so that geometrically, this map wraps the real line around the circle in the complex plane) is a local homeomorphism but not a homeomorphism. The map ${\displaystyle f:S^{1}\to S^{1}}$ defined by ${\displaystyle f(z)=z^{n},}$ where ${\displaystyle n}$ is a fixed integer, wraps the circle around itself ${\displaystyle n}$ times (that is, has winding number ${\displaystyle n}$) and is a local homeomorphism for all non-zero ${\displaystyle n,}$ but it is a homeomorphism only when it is bijective (that is, only when ${\displaystyle n=1}$ or ${\displaystyle n=-1}$).

Generalizing the previous two examples, every covering map is a local homeomorphism; in particular, the universal cover ${\displaystyle p:C\to Y}$ of a space ${\displaystyle Y}$ is a local homeomorphism. In certain situations the converse is true. For example: if ${\displaystyle p:X\to Y}$ is a proper local homeomorphism between two Hausdorff spaces and if ${\displaystyle Y}$ is also locally compact, then ${\displaystyle p}$ is a covering map.

Inclusion maps of open subsets