Lemma: Closed equation subspace

The subspace $P_m\mathcal{D}$ is closed in $\mathcal{D}$. Consequently $Q_m$ is Hausdorff. Convergence in ${\mathcal{D}}_C$ passes through the continuous quotient map.

Proof. The continuous map ${\mathsf{E}}_{\mathrm{A}}-{\mathsf{E}}_{\mathrm{R}}:\mathcal{D}\to C^{\infty }(\hat{M})$ has kernel $P_m\mathcal{D}$. One inclusion follows from (eq:green). Conversely, if ${\mathsf{E}}_{\mathrm{A}}d={\mathsf{E}}_{\mathrm{R}}d=:u$, then

a compact causal diamond. Thus $u\in \mathcal{D}$ and $P_{m}u=d$. The kernel is closed, and the remaining assertions follow from the quotient topology.