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 ?
Best Answer
That is the proof. Why are you asking?! :)