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Hausdorff space (Definition)

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=\varnothing $) 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.

Properties

The following properties are equivalent:
  1. $ X$ is a Hausdorff space.
  2. The set
    $\displaystyle \Delta=\{(x,y)\in X\times X:x=y\} $
    is closed in the product topology of $ X\times X$.
  3. For all $ x\in X$, we have
    $\displaystyle \{x\} = \bigcap \{A : A\subseteq X\ $   closed$\displaystyle ,$   $\displaystyle \mbox{$\exists$ open set}$$\displaystyle \ U\ $   such that$\displaystyle \ x\in U\subseteq A\}. $

Important examples of Hausdorff spaces are metric spaces, manifolds, and topological vector spaces.



"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 $\mathnormal{X}$ is Hausdorff if and only if $\Delta(X)$ 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

Other names:  Hausdorff topological space, T2 space
Also defines:  Hausdorff, Hausdorff topology, T2, T2 topology

Attachments:
a space $\mathnormal{X}$ is Hausdorff if and only if $\Delta(X)$ is closed (Proof) by mathcam
metric spaces are Hausdorff (Proof) by waj
characterization of $T2$ spaces (Theorem) by matte
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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 121 references to this entry.

This is version 19 of Hausdorff space, born on 2002-02-08, modified 2007-06-19.
Object id is 1855, canonical name is T2Space.
Accessed 22204 times total.

Classification:
AMS MSC: 54D10 (General topology :: Fairly general properties :: Lower separation axioms )
Pending Errata and Addenda
None.
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Discussion
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about the entry called "Hausdorff" by mathforever on 2004-10-06 14:14:17
I have actually the following proposal. There is an entry in encyclopedia called "Hausdorff". Since this entry is just linked to a T2 space, I think it reasonable to rename it to "Hausdorff space".

If you type in the Google the following searchs, you'll get the corresponding amount of links, which can be seen as some indication of how frequently they are used:

"T2 space" - 567
"Hausdorff topological space" - 1.710
"Hausdorff space" - 11.600

So, imagine someone find the notion "Hausdorff space" in some article. He goes to PlanetMath, and find no entry for "Hausdorff space". Of cource if he is determined enough, he could look through every entry with the name Hausdorff and finally finds what he needs, but it is better to have a direct link with name "Hausdorff space".
[ reply | up ]
metric space by vitriol on 2003-02-03 21:13:45
perhaps mention that metric spaces are a common example of hausdorff spaces?
[ reply | up ]
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