Corollary 2

— Let E be a Hausdorff weak space, and C a proper convex cone with vertex 0 in E, that is complete for the uniform structure induced by that of E.

(i) For every point a of E, the intersection $C \cap (a - C)$ is compact.

(ii) Let A, B be two closed sets in C. Then $A + B$ is a closed set in C.

(i) The set of the $(x, y) \in C \times C$ such that $x + y = a$ is compact from cor. 1 and from GT, I, § 10.2, th. 1, b). Now this set is also the set of the $(x, a - x)$ for $x \in C \cap (a - C)$, which proves (i).

(ii) If A and B are closed in C, then $A \times B$ is closed in $C \times C$, therefore $A + B$ is closed in C after cor. 1 and GT, I, § 10.1, prop. 1.