Is every open subset of $\mathbb{R}$ uncountable? I was crafting a proof for the theorem that states every open subset of $\mathbb{R}$ can be written as the union of a countable number of disjoint intervals when this question came up. I feel like the answer is yes, but I'm not sure how to go about proving it or whether there is a crazy construction (like the Cantor Set) that creates a countable, open subset of $\mathbb{R}$. Any ideas?
Real Analysis – Is Every Open Subset of $\mathbb{R}$ Uncountable?
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Best Answer
The empty set.
Otherwise, yes. Every open interval is uncountable, so every nonempty open subset of $\Bbb R$ is uncountable.