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# Hausdorff space

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. 3. 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.

Defines:

Hausdorff, Hausdorff topology, T2, T2 topology, T2 axiom

Related:

SeparationAxioms,T1Space,T0Space,T3Space,RegularSpace,MetricSpace, NormalTopologicalSpace, ASpaceMathnormalXIsHausdorffIfAndOnlyIfDeltaXIsClosed, SierpinskiSpace,HausdorffSpaceNotCompletelyHausdorff, Tychonoff, PropertyThatCompactSetsInASpaceAreClosedLies

Synonym:

Hausdorff topological space, T2 space

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

54D10*no label found*

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## Comments

## metric space

perhaps mention that metric spaces are a common example of hausdorff spaces?

## correction not message!

This should be a correction (others take note too!). As a message it is impossible to "resolve" your comment.

Note that there is an "addendum" correction type, so you are free to suggest whatever you'd like to any entry.

apk

## about the entry called "Hausdorff"

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".

## Re: about the entry called "Hausdorff"

In some far distant future PM will hopefully

have a mathematical front page. A very primitive

version is here:

http://planetmath.org/?op=getobj&from=objects&name=OverviewOfTheContentO...

Once everything is in place, I think it will be

much more easier to find things at PM. Searching

for a mathematical concept is very difficult.

## Re: about the entry called "Hausdorff"

There is no need to rename entries. In order to have "Hausdorff space" appear is to add the terms "Hausdorff space" and "Hausdorff topological space" to the "also defines" list of the entry on T_2 spaces. Also, "Hausdorff" should definitely remain on the "also defines list since otherwise the linker won't pick up on stuff like "If $X$ is also Hausdorff, then a stronger form of this theorem apples." There is no problem with having both "Hausdorff" and "Hausdorff space" on the "also defines" list since the linker picks the longest match.

## Re: Re: about the entry called "Hausdorff"

Ok, I completely agree with it. That is what I actually made as a correction to the entry "T2 space", but it was rejected, perhaps because of nonclearly formulated goal. Now I hope it is more clear :)

## Re: about the entry called "Hausdorff"

Would putting "Hausdorff" in Keywords Section serve the same purpose?

## Re: about the entry called "Hausdorff"

First of all, I don't see any Keywords Section. But even if such section would exist, putting "Hausdorff" there, won't help a poor man to find out what is "Hausdorf space".

## Re: about the entry called "Hausdorff"

it would (it'd improve the entry ranked in the search engine)

But again, I don't wee what's the big fuss.

Entries can have SEVERAL titles, that's what the "synonyms" filed are on the editing page.

And mine has both T2 and Hausdorff

and searching both for T2 space and Hausdorff Space will hit mine

f

G -----> H G

p \ /_ ----- ~ f(G)

\ / f ker f

G/ker f

## Re: about the entry called "Hausdorff"

> I don't wee what's the big fuss.

> searching both for T2 space and Hausdorff Space

> will hit mine

Well that's true, the only point is that searching engine doesn't work for the moment on PlanetMath (in fact, as far as I aware it doesn't work for a quite long time). One can make a Google query "Hausdorff space" of the PlanetMath, and than the first thing which will come up will be your entry. Anyway, as I see you already made an entry "Hausdorff Space" ;), and "that is good so" ;)