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Hausdorff space
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(Definition)
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A topological space $(X,\tau)$ is said to be $T_2$ (or said to satisfy the $T_2$ axiom) if given distinct $x,y\in X$ , there exist disjoint open sets $U,V\in\tau$ (that is, $U\cap V=\emptyset$ ) such that $x\in U$ and $y\in V$ .
A $T_2$ space is also known as a Hausdorff space. A Hausdorff topology for a set $X$ is a topology $\tau$ such that $(X,\tau)$ is a Hausdorff space.
The following properties are equivalent:
- $X$ is a Hausdorff space.
- The set $$ \Delta=\{(x,y)\in X\times X:x=y\} $$ is closed in the product topology of $X\times X$ .
- For all $x\in X$ , we have $$ \{x\} = \bigcap \{A : A\subseteq X\ \mbox{closed}, \mbox{$\exists$ open set}\ U\ \mbox{such that}\ x\in U\subseteq A\}. $$
Important examples of Hausdorff spaces are metric spaces, manifolds, and topological vector spaces.
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"Hausdorff space" is owned by yark. [ full author list (3) | owner history (2) ]
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See Also: separation axioms, T1 space, T0 space, T3 space, regular space, metric space, normal, a space  is Hausdorff if and only if  is closed, Sierpinski space, Hausdorff space not completely Hausdorff, Tychonoff space, The property that compact sets in a space are closed lies strictly between T1 and T2, applications of Urysohn's Lemma to locally compact Hausdorff spaces
| Other names: |
Hausdorff topological space, T2 space |
| Also defines: |
Hausdorff, Hausdorff topology, T2, T2 topology, T2 axiom |
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Cross-references: topological vector spaces, manifolds, metric spaces, product topology, closed, equivalent, properties, open sets, disjoint, axiom, topological space
There are 137 references to this entry.
This is version 20 of Hausdorff space, born on 2002-02-08, modified 2008-08-21.
Object id is 1855, canonical name is T2Space.
Accessed 29146 times total.
Classification:
| AMS MSC: | 54D10 (General topology :: Fairly general properties :: Lower separation axioms ) |
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Pending Errata and Addenda
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