absorbing subset is a barrel. The closure of a vector subspace of a TVS is a vector subspace. Every finite dimensional vector subspace of a Hausdorff TVS is closed. The sum of a closed vector subspace and a finite-dimensional vector subspace is closed. If is a vector subspace of and is a closed neighborhood of the origin in such that is closed in then is closed in The sum of a compact set and a closed set is closed. However, the sum of two closed subsets may fail to be closed (see this footnote for examples). If and is a scalar then where if is Hausdorff, then equality holds: In particular, every non-zero scalar multiple of a closed set is closed. If and if is a set of scalars such that neither contain zero then If then is convex. If then and so consequently, if is closed then so is If is a real TVS and then where the left hand side is independent of the topology on moreover, if is a convex neighborhood of the origin then equality holds. For any subset where is any neighborhood basis at the origin for However, and it is possible for this containment to be proper (for example, if and is the rational numbers). It follows that for every neighborhood of the origin in Closed hullsIn a locally convex space, convex hulls of bounded sets are bounded. This is not true for TVSs in general. The closed convex hull of a set is equal to the closure of the convex hull of that set; that is, equal to The closed balanced hull of a set is equal to the closure of the balanced hull of that set; that is, equal to The closed disked hull of a set is equal to the closure of the disked hull of that set; that is, equal to If and the closed convex hull of one of the sets or is compact then If each have a closed convex hull that is compact (that is, and are compact) then Hulls and compactnessIn a general TVS, the closed convex hull of a compact set may to be compact. The balanced hull of a compact (respectively, totally bounded) set has that same property. The convex hull of a finite union of compact sets is again compact and convex. Other propertiesMeager, nowhere dense, and BaireA disk in a TVS is