Talk:Hausdorff space

From Wikipedia, the free encyclopedia

WikiProject Mathematics
Mathematics Portal
This article is within the scope of WikiProject Mathematics, which collaborates on articles related to mathematics.
Mathematics rating: B Class Mid Priority  Field: Topology

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

[edit] 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 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.

[edit] 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 talk 00:06, 3 December 2005 (UTC)