Corollary

— In a topological vector space $E$, every hyperplane is either closed or everywhere dense.

In fact, the closure of a homogeneous hyperplane $H$ can only be $H$ or the whole space $E$, since it is a vector subspace containing $H$ (prop. 1).

A hyperplane $H$ is closed in $E$ if, and only if, $\complement H$ contains an interior point.

The vector subspace $M$ generated by a set $A$, in a topological vector space $E$, is the set of linear combinations of points of $A$ (A, II, § 1.7, prop. 9); the closure of $M$ in $E$ is, by prop. 1, the smallest closed vector subspace containing $A$; we say that this is the closed vector subspace generated by $A$.