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Concept
Local homeomorphism
Summary
In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure.
If is a local homeomorphism, is said to be an étale space over Local homeomorphisms are used in the study of sheaves. Typical examples of local homeomorphisms are covering maps.
A topological space is locally homeomorphic to if every point of has a neighborhood that is homeomorphic to an open subset of
For example, a manifold of dimension is locally homeomorphic to
If there is a local homeomorphism from to then is locally homeomorphic to but the converse is not always true.
For example, the two dimensional sphere, being a manifold, is locally homeomorphic to the plane but there is no local homeomorphism
A function between two topological spaces is called a if for every point there exists an open set containing such that the is open in and the restriction is a homeomorphism (where the respective subspace topologies are used on and on ).
Local homeomorphisms versus homeomorphisms
Every homeomorphism is a local homeomorphism. But a local homeomorphism is a homeomorphism if and only if it is bijective.
A local homeomorphism need not be a homeomorphism. For example, the function defined by (so that geometrically, this map wraps the real line around the circle) is a local homeomorphism but not a homeomorphism.
The map defined by which wraps the circle around itself times (that is, has winding number ), is a local homeomorphism for all non-zero but it is a homeomorphism only when it is bijective (that is, only when or ).
Generalizing the previous two examples, every covering map is a local homeomorphism; in particular, the universal cover of a space is a local homeomorphism.
In certain situations the converse is true. For example: if is a proper local homeomorphism between two Hausdorff spaces and if is also locally compact, then is a covering map.
Official source
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Ontological neighbourhood
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