Corollary 2

Let E be a topological vector space over a non-discrete valued division ring. Every vector subspace D (of dimension 1) which is the algebraic complement of a closed homogeneous hyperplane H is also the topological complement of H.

In D, the set $\{0\}$ is closed, being the intersection of D and the closed set H; D is therefore Hausdorff. But as E/H is also Hausdorff, the canonical mapping of D on E/H, which is linear, is also an isomorphism by prop. 2, from which the conclusion follows (GT, III, § 6.2).