Past exam of the metric spaces and complex analysis course of the University of Oxford / 2014 / 1 / b / iv

True. For any continuous map, $f(\overline{A})\subseteq \overline{f(A)}$: every open neighbourhood of $f(x)$ pulls back to an open neighbourhood of $x\in \overline{A}$, and therefore meets $A$ in its preimage.

Apply the same inclusion to the continuous inverse of the homeomorphism:Applying $f$ gives the reverse inclusion, provingThe closures are taken in their respective ambient spaces.