PlanetMath: locally homeomorphic

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locally homeomorphic

(Definition)

Let and be topological spaces. Then is locally homeomorphic to , if for every $x\in X$ there is a neighbourhood $U\subseteq X$ of and an open set $V\subseteq Y$ , such that and with their respective subspace topology are homeomorphic.

Let $X=\{1\}$ and $Y=\{2,3\}$ 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 $X=\{1\}$ be a discrete space with one element, but now let $Y=\{2,3\}$ the space with topology $\{\emptyset,\{2\},Y\}$ . 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 $Y=\{2,3\}$ be indiscrete. Then neither is locally homeomorphic to nor the other way round.
Non-trivial examples arise with locally Euclidean spaces, especially manifolds.

"locally homeomorphic" is owned by GrafZahl.

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