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locally homeomorphic (Definition)

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

Examples

  • Let $ X=\{1\}$ and $ Y=\{2,3\}$ be discrete spaces with one resp. two elements. Since $ X$ and $ Y$ 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 $ X$ is still locally homeomorphic to $ Y$, but $ Y$ is not locally homeomorphic to $ X$, since the smallest neighbourhood of $ 3$ already has more elements than $ X$.
  • Now, let $ X$ be as in the previous examples, and $ Y=\{2,3\}$ be indiscrete. Then neither $ X$ is locally homeomorphic to $ Y$ 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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See Also: locally Euclidean

Other names:  local homeomorphy
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Cross-references: manifolds, locally Euclidean spaces, cardinalities, discrete spaces, homeomorphic, subspace topology, neighbourhood, topological spaces
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This is version 1 of locally homeomorphic, born on 2005-05-07.
Object id is 7020, canonical name is LocallyHomeomorphic.
Accessed 2838 times total.

Classification:
AMS MSC54-00 (General topology :: General reference works )

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