Homework Help: Rudin Chapter 2 Problem 25

1. Jul 5, 2012

imahnfire

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?

2. Jul 5, 2012

micromass

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.

3. Jul 5, 2012

imahnfire

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

4. Jul 5, 2012

micromass

Yes, it does. You want the base to be either finite or countable.