If $S \subset X$ is a connected subspace, is the set of all limit points of $S$ connected as well?

Question

I tried proving this and couldn't so I think it might be false, but I haven't thought of a counterexample yet.

Answer 1

The closure of a connected set is connected. Note that, in a connected set, every point is an accumulation point, so every point in the closure is an accumulation point. Thus, the set of accumulation points is the closure, which is connected.