It is now April! Did you know that April is one of the months in which every compact metric space is separable?

Proof. Let it be April, and let M be a compact metric space. Because M is compact, it is totally bounded, so for all n∈ℕ, we can cover M with finitely many open balls of radius 1/n. The centers of all such balls are a countable set which we can call C. But C is dense, because an arbitrary point p∈M is a limit point of C: an ε-neighborhood of p must contain the center of one the balls in our covering of M with ε/2-balls. Thus M contains a countable dense subset.