Hausdorff spaces such that every subset is a retract

Question

Let $(X,\tau)$ be a Hausdorff space such that for every non-empty $A\subseteq X$ there is a continuous map $r:X\to A$ such that $r(a) = a$ for all $a\in A$. Does $\tau$ have to be discrete?

Answer 1

Since a retract of a Hausdorff space is closed, such a space must be discrete.