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First comment

I've added a reference to what is, as far as I can find, the first article (Shimrat, 1956) containing a proof that every space can be written as a quotient of a Hausdorff space; this seems to me like a non-trivial fact, so I didn't think it was appropriate to include the statement with no reference or explanation. Some readers might be interested to know that John Isbell generalized this result in A note on complete closure algebras. Math. Systems Theory 4 (1969), although that's probably not worth mentioning in this article.67.85.181.241 (talk) 04:06, 25 August 2009 (UTC)

I've just started trying to learn some topology, and I've come across this definition a few times. While I think I can visualise the specific example - two points, disjoint open sets around them - I don't feel I fully understand it. Can anyone help me (and presumably anyone else new to topology)?

Are there any immediate and more graspable consequences that follow from a topological space being Hausdorff? Why is Hausdorff-ness important? Are most interesting and useful spaces Hausdorff? What do non-Hausdorff spaces look like: are they ugly and weird, are there significant examples that naturally crop up? - Stuart Presnell

This line

  Limits of sequences (when they exist) are unique in Hausdorff spaces.

Is a typical example of the ways in which Hausdorff spaces are 'nice'. --Matthew Woodcraft

Is the contrapositive of this true? If a space is non-Hausdorff, does this mean that the limits of sequences are not unique? -- Stuart Presnell

This is not the contrapositive, it is the converse, and it is false. As to your original question: most topological spaces encountered in analysis are Hausdorff (most of them are even metric spaces, but not all, see e.g. weak topology). An important non-Hausdorff topology is the Zariski topology in algebraic geometry. --AxelBoldt

An example of limit behaviour in a non-Hausdorff space:
Let X = { 1, 2 } and T = { Ø , X }
T is then a topology on X (called the chaotic topology).
The sequence 1,1,1,1,1... has both 1 and 2 as limits, basically because the topology is incapable of distinguising between them.
A non-Hausdorff space will always have at least one pair of indistinguishable points, so a sequence with more than one limit can be constructed as above. -- Tarquin

I'm pretty sure that one can construct some non-first-countable non-Hausdorff T₁ space where limits of sequences are unique. I think Hausdorff spaces can be characterized by the fact that limits of filters are unique. --AxelBoldt

"particularly nice" - Zoe

A not-so-nice property of non-Hausdorff spaces

If a space X carries a non-Hausdorff topology, it is impossible for continuous functions with values in the real or complex numbers (or any Hausdorff space Y, for that matter) to separate points. A function ${\displaystyle f:X\rightarrow Y}$ is said to separate the points x and y if ${\displaystyle f(x)\neq f(y)}$. Assume x and y are non-Hausdorff points, i.e. for any two open sets A and B with ${\displaystyle x\in A,y\in B}$ we have ${\displaystyle A\cap B\neq \emptyset }$. Let Y be a Hausdorff space and ${\displaystyle f:X\rightarrow Y}$ a function separating x and y. Since Y is Hausdorff, there exist disjoint open sets U and V with ${\displaystyle f(x)\in U,f(y)\in V}$. Would f be continuous then ${\displaystyle A=f^{-1}(U)}$ and ${\displaystyle B=f^{-1}(V)}$ would be disjoint open sets with ${\displaystyle x\in A,y\in B}$. But this is impossible, so f cannot be continuous.

The importance of this fact is that non-Hausdorff spaces cannot be adequately described by continuous functions on them.

That could just as well be an argument for why Hausdorff spaces are not "nice"! (whatever that means...) Thehotelambush (talk) 09:10, 30 March 2009 (UTC)

Hausdorff redirect

I feel like the word Hausdorff is used far more often to refer to a property of spaces than it is to refer to Mr. Felix Q Hausdorff. Accordingly, I think Hausdorff should redirect here, rather than there. -lethe 00:06, 3 December 2005 (UTC)

Closed singletons iff T1, not iff T2

In the article it was stated:

For a topological space X, the following are equivalent:

• X is Hausdorff space.
• Every singleton set contained in X is equal to the intersection of all closed neighbourhoods containing it.

This is not true. A set is closed iff it is equal to it's closure, which is the intersection of all closed sets containing the set. Therefore the second condition above means "every singleton is closed". But this is a condition equivalent to the given space being T1. As T1 spaces exist which are not T2, the stated equivalence does not hold.

Therefore I removed from the article the following line:

• Every singleton set contained in X is equal to the intersection of all closed neighbourhoods containing it.

--87.205.171.230 (talk) 13:03, 14 August 2008 (UTC)

I've restored it, as it's correct. The intersection of the closed neighbourhoods of a point is not the same thing as the intersection of the closed sets containing the point. --Zundark (talk) 13:45, 14 August 2008 (UTC)

Citation problem

The citation for the statement "in a Hausdorff space every pair of disjoint compact sets can also be separated by neighborhoods" which currently links to http://planetmath.org/?op=getobj&from=objects&id=4193 does not prove the quoted statement. It proves a weaker statement that a point and a compact set in a hausdorff space can be separated by neighborhoods. I have been able to prove the quoted statement based on the ideas from the planetmath proof. I can supply the proof if desired, however I am sure that someone else on here could do it as well. — Preceding unsigned comment added by 152.14.226.95 (talk) 20:49, 1 August 2011 (UTC)

I've changed the reference (now p124 of Willard). --Zundark (talk) 21:04, 1 August 2011 (UTC)

Equivalence of Hausdorf and T2

Some authors (Munkres for example) define T2 as a weaker condition than Hausdorf: the T2 axiom states that given any two points, each has a neighborhood that does not contain the other (i.e. not necessarily disjoint).

Is it worth pointing this out in the article or is it hair splitting? Has the topological world moved on from 1975 when Munkres was published? Mr. Swordfish (talk) 02:19, 3 February 2021 (UTC)

Nevermind. Misreading Munkres. That's T1 not T2. Mr. Swordfish (talk) 18:27, 3 February 2021 (UTC)

"Non-Hausdorff" listed at Redirects for discussion

An editor has identified a potential problem with the redirect Non-Hausdorff and has thus listed it for discussion. This discussion will occur at Misplaced Pages:Redirects for discussion/Log/2022 January 27#Non-Hausdorff until a consensus is reached, and readers of this page are welcome to contribute to the discussion. Jay (talk) 17:21, 3 February 2022 (UTC)

"R1 space" listed at Redirects for discussion

An editor has identified a potential problem with the redirect R1 space and has thus listed it for discussion. This discussion will occur at Misplaced Pages:Redirects for discussion/Log/2022 April 22#R1 space until a consensus is reached, and readers of this page are welcome to contribute to the discussion. 1234qwer1234qwer4 13:01, 22 April 2022 (UTC)

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