| Revision as of 01:18, 14 January 2002 editAxelBoldt (talk | contribs)Administrators44,499 editsNo edit summary← Previous edit | Revision as of 00:39, 21 January 2002 edit undoTarquin (talk | contribs)14,993 editsNo edit summaryNext edit → | ||
| Line 26: | Line 26: | ||
| 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 non-Hausdorff space:<br> | |||
| Let X = { 1, 2 } and T = { empty set, X } <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> | |||
| A non-Hausdorff topology will always have at least one pair of indistinguishable points, so a sequence with more than one limit can be constructed as above. -- Tarquin | |||
Revision as of 00:39, 21 January 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?
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'.
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 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 non-Hausdorff space:
Let X = { 1, 2 } and T = { empty set, 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 topology will always have at least one pair of indistinguishable points, so a sequence with more than one limit can be constructed as above. -- Tarquin