Maybe I've just got a wrong idea of what's going on here, or maybe I just have a definition wrong, but this theorem seems a bit difficult for me to understand, and I think I just need an example. For instance, consider the open set (0,1). This theorem seems like it's saying I can write (0,1)=(0,1/2)U(1/2+epsilon,1) or something? I'm just a bit confused as to how one would concretely form an open set out of disjoint open intervals I guess.
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No, $(0,1)$ is exactly itself. – Kenny Lau Sep 14 '17 at 15:18
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Related – Kenny Lau Sep 14 '17 at 15:18
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A set is countable if it has finitely many elements or if its elements can be put into a one-to-one correspondence with the counting numbers. So the interval $(0,1)$ indeed consists of a countable union of disjoint open sets, namely one such set. The theorem is based upon the fundamental idea that the reals have a countable dense subset, namely the rationals. Then the set of $\epsilon$-balls that are centered on the rational numbers and have rational radii is a countable collection of sets. Any open set in the reals is a union of some of these open sets.
Eric Fisher
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+1. It is worth noting that some texts do (sadly) use "countable" to mean "countable and infinite," so the OP's confusion is understandable. – Noah Schweber Sep 14 '17 at 19:06
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Indeed, you're correct. Even an excellent mathematician may occasionally lapse into ambiguity, but mathematics itself is so beautifully clear. – Eric Fisher Sep 15 '17 at 01:26