Why are open sets in $\mathbb{R}$ uncountable?
We've proven that $\mathbb{R}$ is uncountable and our definition for the open set A is $x \in A \implies \forall \epsilon: U_\epsilon(x) \cap A \neq \emptyset$
Why are open sets in $\mathbb{R}$ uncountable?
We've proven that $\mathbb{R}$ is uncountable and our definition for the open set A is $x \in A \implies \forall \epsilon: U_\epsilon(x) \cap A \neq \emptyset$
Every non-empty open set in $\Bbb R$ contains an open interval. Given an open interval $O$, there is a bijection from $O$ to $(0,1)$ (or $\Bbb R$; use the inverse tangent function appropriately altered), which is uncountable.
And one from $(a,b)$, with $0
https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
Consider the above article. Historically the uncountability of intervals was actually used to prove it for the real numbers. Once you have it for (0,1) the order-preserving map shows it for each interval. It is trivial to extend this to unions of intervals.