Let $(X,\tau)$ be a topological space. A retraction is a continuous map $r:X\to X$ such that $r$ is the identity on $\text{im}(r)$. We call $S\subseteq X$ a retract of $X$ if there is a retraction $r:X\to X$ such that $\text{im}(r) = S$.

We say that $(X,\tau)$ is $rc$ if all retracts are closed. It turns out that all Hausdorff spaces have this property.

Is the class of $rc$-spaces closed under topological products?

Let $(X,\tau)$ be a topological space. A retraction is a continuous map $r:X\to X$ such that $r$ is the identity on $\text{im}(r)$. We call $S\subseteq X$ a retract of $X$ if there is a retraction $r:X\to X$ such that $\text{im}(r) = S$.

We say that $(X,\tau)$ is rc if all retracts are closed. It turns out that all Hausdorff spaces have this property.

Is the class of rc-spaces closed under topological products?