Thread: Connected Hausdorff Spaces

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

Originally Posted by Turloughmack

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

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

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

Originally Posted by Turloughmack

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

The link I gave you brings an example of such a space in theorem 1, first page!

Tonio

Originally Posted by Turloughmack

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

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.

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?