# 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.