Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
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
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.