Principles of Mathematical Analysis, Rudin, 3th ed, Chapter 2, Problem 24

Problem

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

Answer

For $n \in \mathbb{N}$, choose sequentially $x_i \in X$ for $i=1,2,\cdots$ satisfying $d(x_i,x_j) \geq 1/n$ for $j=1,\cdots,i-1$. Then, this process must stop after a finite number of steps (If this process does not stop, then it contradicts to the fact that every infinite subset has a limit point). Define $S_n$ be the set of such points. Note that $S:=\bigcup_{n=1}^\infty S_n$ is countable. It remains to show that $S$ is dense in $X$. Fix $x \in X$ and $\epsilon>0$. We can choose $n \in \mathbb{N}$ such that $1/n<\epsilon$. Then $x \in N_{1/n}(x_\ast)$ for some $x_\ast \in S_n$, which implies $x_\ast \in N_\epsilon(x)$.

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