Lemma: The inherited scalar topology

The subspace $P_m\mathcal{D}$ is closed in $\mathcal{D}$. Consequently $Q_m$ is Hausdorff and the map ${\mathcal{D}}_{C_f}\to Q_m$ is continuous.

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 ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}a={{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}a=u$, then $u$ has support in the compact causal diamond of $\mathrm{supp}a$. Hence $u\in \mathcal{D}$ and $P_{m}u=a$. The converse follows from the Green identities.