Can anybody tell me why it is so hard to find why countable connected hausdorff spaces?

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- November 16th 2010, 04:31 AMTurloughmackConnected Hausdorff Spaces
Can anybody tell me why it is so hard to find why countable connected hausdorff spaces?

- November 16th 2010, 04:39 AMtonio

The "why" is because maths is that way. Now look here JSTOR: An Error Occurred Setting Your User Cookie to find an example and links to others.

Tonio - November 16th 2010, 06:38 AMTurloughmack
Can you give me a direct definition or example of a countable connected Hausdorff space?

None of the papers I look at do, they all just mention that Urysohn constructed the countable connected Hausdorff space but they don't show the definition.

Please Help - November 16th 2010, 07:16 AMtonio
- November 16th 2010, 06:21 PMDrexel28
Just a comment: half of the reason why it's so hard is that the space can't be metrizable (countable metric spaces [with more than one point] are always disconnected) and people usually think in terms of metric spaces, like it or not.

P.S. I assume you mean countably infinite, otherwise any one-point discrete space is countable Hausdorff and connected. - November 18th 2010, 06:20 AMTurloughmack
Can anybody tell me what is a countable connected Hausdorff space??

I know that

Connected: A topological space is connected if it is not the union of a pair of disjoint sets

Countable: A set is countable if it is finite or countably infinite

Hausdorff space: has distinct points which have disjoint neighborhoods.

So by applying these three definitions does that mean a countable connected Hausdorff space is a finite connected space with disjoint neighborhoods?

How would one picture this?