Talk:Hausdorff space

message: 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)
thread_title:  First comment 

message: 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  f:X\rightarrow Y is said to separate the points x and y if f(x)\neq f(y). Assume x and y are non-Hausdorff points, i.e. for any two open sets A and B with x\in A, y\in B we have A\cap B\neq\emptyset. Let Y be a Hausdorff space and f:X\rightarrow Y a function separating x and y. Since Y is Hausdorff, there exist disjoint open sets U and V with f(x)\in U, f(y)\in V. Would f be continuous then A=f^{-1}(U) and B=f^{-1}(V) would be disjoint open sets with 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)
thread_title:  A not-so-nice property of non-Hausdorff spaces

message: 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 talk 00:06, 3 December 2005 (UTC)
thread_title:  Hausdorff redirect 

message: 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)
replies: text: 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)
thread_title:  Closed singletons iff T1, not iff T2 

message: The citation for the statement "in a Hausdorff space every pair of disjoint compact sets can also be separated by neighborhoods[8]" 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)
replies: text: I've changed the reference (now p124 of Willard). --Zundark (talk) 21:04, 1 August 2011 (UTC)
thread_title:  Citation problem 

message: 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)
replies: text: Nevermind.  Misreading Munkres.  That's T1 not T2. Mr. Swordfish (talk) 18:27, 3 February 2021 (UTC)
thread_title:  Equivalence of Hausdorf and T2 

message: An editor has identified a potential problem with the redirect :Non-Hausdorff and has thus listed it for discussion. This discussion will occur at Wikipedia: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)
thread_title:  "Non-Hausdorff" listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]] 

message: An editor has identified a potential problem with the redirect :R1 space and has thus listed it for discussion. This discussion will occur at Wikipedia: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)
thread_title:  "R1 space" listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]]