This is an assignment for class so, please don't tell me how to do this problem or give too much away. I just need to know why I need to prove this. Let me explain. Here is the problem from Rudin. Chapter 2, 25 Prove that every compact metric space K has a countable base, and that K is therefore separable. Hint: For every positive integer n, there are finitely many neighborhoods, radius 1/n whose union covers K. Relevant Theorems: 1. Every separable metric space has a countable base. 2. If X is a metric space in which every infinite subset has a limit point then X is separable. 3. If E is an infinite subset of a compact set K, then E has a limit point in K. My response: (ignoring the hint) K is compact, so every infinite subset of K has a limit point. Hence, K is separable. Every separable metric space has a countable base. This can't be right. Why did Rudin give the "hint?"