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neighbourhood [2014/10/25 14:47] nikolaj old revision restored (2014/10/25 14:22) |
neighbourhood [2014/10/25 14:47] nikolaj old revision restored (2014/10/25 14:25) |
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| === Predicates === | === Predicates === | ||
| Consider $X$ together with a topology, then | Consider $X$ together with a topology, then | ||
| - | |||
| - | //Hausdorff space// means you can separate Neighbourhoods: | ||
| - | |||
| - | | @#EEEE55: predicate | @#EEEE55: $X$ ... Hausdorff space $ \equiv \forall (x,y\in X).\ x\neq y \implies \exists (U_x\in\mathrm{Neighbourhood}(x), U_y\in \mathrm{Neighbourhood}(y)).\ U_x\cap U_y=\emptyset $ | | ||
| //locally euclidean space// means $X$ is homeomorphic to $\mathbb R^n$: | //locally euclidean space// means $X$ is homeomorphic to $\mathbb R^n$: | ||
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| === Reference === | === Reference === | ||
| Wikipedia: | Wikipedia: | ||
| - | [[https://en.wikipedia.org/wiki/Neighbourhood_%28mathematics%29|Neighbourhood]], | ||
| [[http://en.wikipedia.org/wiki/Hausdorff_space|Hausdorff space]], | [[http://en.wikipedia.org/wiki/Hausdorff_space|Hausdorff space]], | ||
| [[http://en.wikipedia.org/wiki/Topological_manifold|Topological manifold]] | [[http://en.wikipedia.org/wiki/Topological_manifold|Topological manifold]] | ||
