TITLE: Is every open subset of $ \mathbb{R} $ uncountable?
QUESTION [21 upvotes]: 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?

REPLY [6 votes]: I assume you don't want a proof but hints. To prove all intervals are uncountable you could first try proving (0,1) is uncountable. Can you construct a decimal fraction of a number that is in (0,1) but isn't in a supposed sequence of all numbers in (0,1)?
After you've established (0,1) is uncountable you could try constructing a bijection between (a,b) and (0,1) and thus proving (a,b) is uncountable.