Is any countable subset of an uncountable set closed

Question

Consider uncountable nonempty $X\subset\mathbb{R}^k$. I wonder if I choose $E\subset X$ such that $E={p_1,p_2,…}$ countable and infinite, then $E$ must be closed in X.
Also, if I choose such set in spaces other than Euclidean space, can I say any countable subset of an uncountable set must be closed?

Answer 1

Not in general, no. For example, consider

$$ \left\{\frac{1}{n} : n \in \mathbb{N}\right\} \subset \mathbb{R}. $$