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.
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.
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?