Constrained data and topology

Write

Put $\bar{A}=\mathsf{r}A$, $\bar{B}=\mathsf{r}B$. Let ${\mathcal{X}}_{r_{\ast }}$ be the smooth tuples supported in ${\overline{B}}_{r_{\ast }}$ satisfying

Give ${\mathcal{X}}_{r_{\ast }}$ its inherited fixed-support smooth Fréchet topology.

Let ${\mathcal{S}}_{r_{\ast }}$ be the solutions of (eq:C1) supported in ${\mathcal{C}}_{r_{\ast }}$, with compact-open smooth topology on the auxiliary slab. Support and differential equations are closed conditions.