Lemma: Closed equation subspace

The subspace $P_m\mathcal{D}$ is closed in $\mathcal{D}$. Hence $Q_m$ is Hausdorff, and convergence in ${\mathcal{D}}_C$ passes to its continuous quotient map.

Proof. The continuous map ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}-{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}:\mathcal{D}\to C^{\infty }(\hat{M})$ has kernel $P_m\mathcal{D}$. If ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}a={{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}a=u$, its support lies in the compact causal diamond $J^{-}(\mathrm{supp}a)\cap J^{+}(\mathrm{supp}a)$. Thus $u\in \mathcal{D}$ and $P_{m}u=a$. The other inclusion follows from the Green identities.