# Rudin Chapter 2 Problem 25

Rudin's problem asks: Prove that every compact metric space K has a countable base.

My concern is how valid this statement really is. Wouldn't a finite compact metric space be unable to have a countable base?

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Rudin probably defines countable as either finite or in bijection with $\mathbb{N}$. So finite things are countable to him.

You should look up his definition to be sure.

Rudin distinguishes finite sets and countable sets in his book. Would this affect the validity of the statement?

Rudin distinguishes finite sets and countable sets in his book. Would this affect the validity of the statement?
Yes, it does. You want the base to be either finite or countable.