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In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local structure. If f : X → Y {\displaystyle f:X\to Y} is a local homeomorphism, X {\displaystyle X} is said to be an étale space over Y . {\displaystyle Y.} Local homeomorphisms are used in the study of sheaves. Typical examples of local homeomorphisms are covering maps.
A topological space X {\displaystyle X} is locally homeomorphic to Y {\displaystyle Y} if every point of X {\displaystyle X} has a neighborhood that is homeomorphic to an open subset of Y {\displaystyle Y}. For example, a manifold of dimension n {\displaystyle n} is locally homeomorphic to R n . {\displaystyle \mathbb {R} ^{n}.}
If there is a local homeomorphism from X {\displaystyle X} to Y , {\displaystyle Y,} then X {\displaystyle X} is locally homeomorphic to Y , {\displaystyle 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 S 2 → R 2 {\displaystyle S^{2}\to \mathbb {R} ^{2}}.