locally homeomorphic
Let and be topological spaces. Then is locally homeomorphic to , if for every there is a neighbourhood of and an http://planetmath.org/node/380open set , such that and with their respective subspace topology are homeomorphic.
Examples
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Let and be discrete spaces with one resp. two elements. Since and have different cardinalities, they cannot be homeomorphic. They are, however, locally homeomorphic to each other.
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Again, let be a discrete space with one element, but now let the space with topology . Then is still locally homeomorphic to , but is not locally homeomorphic to , since the smallest neighbourhood of already has more elements than .
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Now, let be as in the previous examples, and be http://planetmath.org/node/3120indiscrete. Then neither is locally homeomorphic to nor the other way round.
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Non-trivial examples arise with locally Euclidean spaces, especially manifolds.
Title | locally homeomorphic |
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Canonical name | LocallyHomeomorphic |
Date of creation | 2013-03-22 15:14:34 |
Last modified on | 2013-03-22 15:14:34 |
Owner | GrafZahl (9234) |
Last modified by | GrafZahl (9234) |
Numerical id | 4 |
Author | GrafZahl (9234) |
Entry type | Definition |
Classification | msc 54-00 |
Synonym | local homeomorphy |
Related topic | LocallyEuclidean |