Proposition: Hausdorffness and differential continuity

The topology ${\tau }_{\Sigma }$ is Hausdorff. Every fixed-direction operator $D_{\eta }$ and conjugation are continuous. Evaluation of $F$, or any fixed directional descendant, at any fixed smooth configuration is continuous.

Proof. Lemma lem:unique makes the coefficient inclusion injective into a Hausdorff space, so (eq:graph) is Hausdorff.

Contraction of a compact distribution with a fixed smooth field is strongly continuous: insertion of that field maps bounded sets of smooth target tests to bounded sets of source tests. The finite degree shifts are continuous on the locally convex direct sum. The same argument handles fixed evaluations.

For the boundary coordinates,

Thus differentiation shifts coordinates already present in the topology. Conjugation replaces the configuration and direction list by their conjugates and conjugates the trace.