Does there exist a countable metric space which is connected? If so, give an example andProblem:proveit's in fact connected.

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- May 17th 2010, 08:14 PMDrexel28Easy Metric Space Topology
Does there exist a countable metric space which is connected? If so, give an example and__Problem:____prove__it's in fact connected. - May 18th 2010, 01:45 PMBruno J.
How about the metric space which contains only one point? (Giggle)

- May 18th 2010, 04:23 PMDrexel28
- May 19th 2010, 10:52 AMOpalg
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. - May 19th 2010, 12:52 PMDrexel28
Nice!

Mine:

__Spoiler__:

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.