April Is Separability Month

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.

One thought on “April Is Separability Month

  1. Thanks for reminding me why I love April.

    I can't wait for all the theorems I'll be able to prove again this year unconditionally, only about a month left now! Truly, April is a brief crack of light between two eternities of light.

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