Let $(X,\tau)$ be a topological 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 this imply that $\tau$ is either discrete ($\tau = \mathcal{P}(X)$) or indiscrete ($\tau = \{\emptyset, X\}$)?