Is there a separable metric space with an uncountable number of clopen sets?

Question

I can't seem to think of one. I'm taking a basic course in descriptive set theory, and I still don't have a very strong intuition for these concepts. As I think about it clopen sets are those which are separated from the rest, as in the discrete topology all sets are clopen. Also, there is a relation between clopen sets and connectedness. On the other hand being separable means that you can find a countable set which gets as close to any element as you want. This problem seems to go against my way of thinking about it because there has to be an uncountable number of separated sets but whose points stay close to a countable set. I was wondering if anyone has a better intuition for these two concepts and knows of an example which illustrates this.

Answer 1

Yes: $\Bbb N$ with the discrete metric. Every set in that space is clopen, and every countable space is of course separable.

Less trivially, let $X$ be the irrationals with the usual topology. For $n\in\Bbb N$ let $p_n=n+\frac13$ and $q_n=n+\frac23$. For each infinite sequence $\sigma=\langle b_n:n\in\Bbb N\rangle$ of zeroes and ones let

$$U_\sigma=\bigcup\big\{(p_n,q_n)\cap X:b_n=1\big\}\,;$$

there are uncountably many (in fact $2^\omega=\mathfrak{c}$) such sets, and each of them is clopen in $X$.