# Thread: countable connected Hausdorff space

1. ## countable connected Hausdorff space

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?

2. Originally Posted by Turloughmack
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?

Anyone learning topology should already know that not everything in maths is easy, or even possible, to "picture" in a standard way.

A countable connected Hausdorf space is a (topological) space which is Hausdorrf, connected and countable...what else do

you need?! A photo? I don't think there is one, although perhaps there exist somewhere some more "intuitive" descripition of this.

Perhaps in the book "Counterexamples in Topology", by Steen and Seebach, there's something...

Tonio