How may limit points can a countable space have?

Question

I can think of a countable topological space with a finite number of limit points. I was wondering if the set of limit points could also be infinite.

To make it more clear, I'm asking if there is a countable topological space $X$ with a infinite subset $A$ such that every $x\in A$ is a limit point of $X$.

Answer 1

Take Q, the space of rational numbers. Every point is a limit point. So there's a countable set of limit points.