Problem: Does there exist a countable metric space which is connected? If so, give an example and prove it's in fact connected.
Choose . If it is an isolated point then is open and closed, so is not connected. Otherwise, let . Then D is a countable set of positive real numbers, with . It may happen that D is not bounded above, in which case define In any case, we can find a real number with Let , . Then are disjoint nonempty open subsets of with . So cannot be connected.
It is essential for that argument that should be a metric space. This example shows that a countable Hausdorff topological space can be connected.
And while I was aware of the other result you quoted there is something that can be said even more specifically for countable metric spaces. Namely if is a countable metric space without an isolated point then with the obvious metric. From there the conclusion would follow. It's a nice proof too.