I understand this proof of the theorem, but my problem is applying this to some open sets. For example, how would be the disjoint open sets in the countable union that cover, for instance, the open sets $\mathbb{R}$, $(0,1)$ and $(0,1) \cup (2,3)$ of $\mathbb{R}$ in the context of this proof? I'm having a really hard time visualizing them. Thanks.
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3Note that sets of cardinality one or two are also countable – Hagen von Eitzen Sep 24 '18 at 17:35
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but then would't the union be uncountable? – Sep 24 '18 at 17:46
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2Countable in this context means "finite or countably infinite." So (0,1) is already a finite union of open intervals. – user4894 Sep 24 '18 at 17:57
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Oh, yes, now I see. Ty. But what about $\mathbb{R}$? – Sep 24 '18 at 17:58
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2$\mathbb{R}$ is an open set so it covers itself with a single open set. – CyclotomicField Sep 24 '18 at 18:04
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Oh, of course. I'm feeling very dumb right now. Thank you very much. – Sep 24 '18 at 18:10
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1No worries, I've seen a lot of people get tripped up on this one. – user4894 Sep 24 '18 at 18:31
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If you write these comments as an answer I can give you the points – Sep 26 '18 at 13:11