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.
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.
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 .
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.
|Date of creation||2013-03-22 15:14:34|
|Last modified on||2013-03-22 15:14:34|
|Last modified by||GrafZahl (9234)|