# Baby Rudin Exercise 2.24

I have some difficulties solving the following exercise (Baby Rudin 2.24)

Let $X$ be a metric space in which every infinite subset has a limit point. Prove that $X$ is separable.

In order to solve this I try to find a connected set which contains an infinite subset that has no limit point.

So, the interval $[0 \dots 1]$ is a connected set, but it is a compact one also.

According to Baby Rudin theorem 2.37 any infinte subset $E$ of a compact set $K$ has a limit point in $K$.

It appears that I can't find a subset to contradict exercise 2.24.

I know that there is a hint in the book, I try not to read it.

Thnks

• Separability has nothing to do with connectedness. You need to show that $X$ has a countable dense subset. – Daniel Fischer Sep 4 '15 at 18:40
• Well' I thought that a set has two choices, being connected or seperated. – user251106 Sep 4 '15 at 18:56
• It's a different thing. The words are similar, but mean something completely different. (Also, "separated" often is used as a synonym of Hausdorff.) – Daniel Fischer Sep 4 '15 at 18:58