A countable subspace of an uncountable metrizable space is totally disconnected

Question

Let $(X,\tau) $ be a metrizable topological space with $X$ being uncountable; show any countable subspace of $X$ is totally disconnected. I'm not sure how to approach this problem or that's a true statement.

Answer 1

If $X$ is $T_4$ (so metric spaces work), and connected and has at least 2 points, then $X$ is has size at least $\mathfrak{c}$, so certainly uncountable.

Proof: let $p \neq q$ in $X$. Then $\{p\}$ and $\{q\}$ are closed disjoint, so by Urysohn's lemma there exists a continuous $f: X \rightarrow [0,1]$ with $f(p) = 0 $ and $f(q) = 1$. So $f[X]$ is connected (continuous image of a connected space) and it contains $0$ and $1$ so $f[X] = [0,1]$ (connected sets in the reals are order convex), which by standard set theory implies $\mathfrak{c} \le \left|X\right|$.

Corollary: if $X$ is hereditarily normal ($T_5$) and has size less than $\mathfrak{c}$ it is totally disconnected: every subset of size $\ge 2$ must have size at least $\mathfrak{c}$, which cannot be, as $X$ is too small.