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Working in $R_{\text usual}$ Topology: Show that there is no uncountable collection of pairwise disjoint open subsets of $\mathbb R$.

Definition of $R_{\text usual}$ I'm working with: $\{U \subseteq \mathbb R: \forall x \in U \exists \ \epsilon \ s.t (x-\epsilon, x+\epsilon) \subseteq U \}$ We are looking for an uncountable collection of pairwise disjoint open subsets of $\mathbb R$,

1.Since rationals are a countable subset of $\mathbb R$, and

2.Every open set in $R_{\text usual}$ also contains rationals (choose an interval of width $2\epsilon$ around x, choose integer n larger that $\frac{1}{2\epsilon}$, choose integer a s.t $\frac{a}{n} \leq (x - \epsilon$) then $\frac{a + 1}{n} \in (x - \epsilon, x + \epsilon)$ by definition of n. )

Then by 1 and 2 there can not exist an uncountable collection of pairwise disjoint sets in $\mathbb R$

Am I in the right direction ?

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  • $\begingroup$ You forgot "open" in your title. $\endgroup$ – zhw. May 17 '16 at 23:23
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    $\begingroup$ @jia So if each open set in this uncountable collection of pairwise disjoint open subsets of $\mathbb{R}$ contains a rational number, how many rational numbers would that be altogether? Do you even need to worry about $\epsilon$? $\endgroup$ – John Wayland Bales May 17 '16 at 23:40
  • $\begingroup$ You mean "I'm working with: $\{U\subseteq \Bbb R : ...\}$ (inclusion not membership). $\endgroup$ – BrianO May 18 '16 at 0:26
  • $\begingroup$ Thank you Brian, I edited :) $\endgroup$ – ಠ_ಠ May 18 '16 at 0:32
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That is the proof. Why are you asking?! :)

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Suppose you have an uncountable collection of open intervals in $\mathbb{R}$. Since $\mathbb{Q}$ is dense in $\mathbb{R}$, you can choose a rational out of each of these intervals. By disjointness, this map is 1-1. We have an injective map of an uncountable set into a countable one. OOPS. Contradiction. By reductio ad absurdum, such a collection cannot exist.

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I think that the direction to look at is to consider that any open set in $\mathbb R$ contains an interval. So: does exists an uncountable collection of disjoint intervals? (But the rational number are countable...)

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