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That's a problem I proved (quite a while back) in tiny Rudin. However, I don't really get it. The other questions were actually useful results - I don't think I've ever come near using this result. Surely it's going to be close to apparent that you're working in an uncountable set?

For instance, examples where this result could be applied but it is hard otherwise to tell that the space is uncountable?

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    I edited the title because the old one sounded to me like you were talking about metric spaces with 2 connected points.2012-01-05
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    @JonasMeyer "Are you asking for examples where this result could be applied but it is hard otherwise to tell that the space is uncountable?" This is how I interpret the question.2012-01-05
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    "Are you asking for examples where this result could be applied but it is hard otherwise to tell that the space is uncountable?" Yes apologies if it was not apparent. I should of maybe said easily proved; I wouldn't call it trivial - though I certainly didn't prove it by showing every connected metric space with two points is uncountable!2012-01-05
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    Well, for instance this leads to a proof of the uncountability of $\mathbb{R}$ (not necessarily the easiest, but a nice enough one, I think). In general I think I agree that this result is not something you are going to use to prove some other result: neither is Fermat's Last Theorem, by the way. Not all results in mathematics have to be directly useful to be interesting.2012-01-05
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    The result says that aside from a trivial one-point topology, there is no countable connected metric space. The "application" of this theorem is not to prove that a set is uncountable, but to tell everyone to stop looking for countable connected metric spaces.2012-01-05
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    Just a small addition: there are countable connected spaces (e.g. the integers with the cofinite topology). There are even [countable connected Hausdorff spaces](http://math.stackexchange.com/a/16673) (the first example of which is due to Urysohn). See also [this MO thread](http://mathoverflow.net/questions/46986/).2012-01-05

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