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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

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  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