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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)? | 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? | 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? | ||
| ⚫ | - ] | ||
| ⚫ | - ] | ||
| This line | This line | ||
| Limits of sequences (when they exist) are unique in Hausdorff spaces. | Limits of sequences (when they exist) are unique in Hausdorff spaces. | ||
| Is a typical example of the ways in which Hausdorff spaces are 'nice'. | Is a typical example of the ways in which Hausdorff spaces are 'nice'. | ||
| ⚫ | --] | ||
| ⚫ | --] | ||
| Is the contrapositive of this true? If a space is non-Hausdorff, does this mean that the limits of sequences are not unique? | Is the contrapositive of this true? If a space is non-Hausdorff, does this mean that the limits of sequences are not unique? | ||
| ⚫ | -- ] | ||
| ⚫ | This is not the ], it is the converse, and it is false. As to your original question: most topological spaces encountered in ] are Hausdorff (most of them are even ], but not all, see e.g. ]). An important non-Hausdorff topology is the Zariski topology in ]. --AxelBoldt | ||
| ⚫ | -- ] | ||
| ⚫ | This is not the ], it is the converse, and it is false. As to your original question: most topological spaces encountered in ] are Hausdorff (most of them are even ], but not all, see e.g. ]). An important non-Hausdorff topology is the Zariski topology in ]. --AxelBoldt | ||
| An example of limit behaviour in a non-Hausdorff space:<br> | An example of limit behaviour in a non-Hausdorff space:<br> | ||
| Let X = { 1, 2 } and T = { Ø , X } <br> | Let X = { 1, 2 } and T = { Ø , X } <br> | ||
| T is then a topology on X (called the chaotic topology).<br> | T is then a topology on X (called the chaotic topology).<br> | ||
| The sequence 1,1,1,1,1... has both 1 and 2 as limits, basically because the topology is incapable of distinguising between them.<br> | The sequence 1,1,1,1,1... has both 1 and 2 as limits, basically because the topology is incapable of distinguising between them.<br> | ||
| 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 | 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-] non-Hausdorff T<sub>1</sub> space where limits of sequences are unique. I think Hausdorff spaces can be characterized by the fact that limits of ''filters'' are unique. --AxelBoldt | |||
Revision as of 15:51, 25 February 2002
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