then there exist subsets that are both but in ; for instance, this will be true of any non-empty proper subset of If is compact, then and this set is compact. Thus the closure of a compact subset of a TVS is compact (said differently, all compact sets are relatively compact), which is not guaranteed for arbitrary non-Hausdorff topological spaces. For every subset and consequently, if is open or closed in then (so that this open closed subsets can be described as a "tube" whose vertical side is the vector space ). For any subset of this TVS the following are equivalent: is totally bounded. is totally bounded. is totally bounded. The image if under the canonical quotient map is totally bounded. If is a vector subspace of a TVS then is Hausdorff if and only if is closed in Moreover, the quotient map is always a closed map onto the (necessarily) Hausdorff TVS. Every vector subspace of that is an algebraic complement of (that is, a vector subspace that satisfies and ) is a topological complement of Consequently, if is an algebraic complement of in then the addition map defined by is a TVS-isomorphism, where is necessarily Hausdorff and has the indiscrete topology. Moreover, if is a Hausdorff completion of then is a completion of Closed and compact setsCompact and totally bounded setsA subset of a TVS is compact if and only if it is complete and totally bounded. Thus, in a complete topological vector space, a closed and totally bounded subset is compact. A subset of a TVS is totally bounded if and only if is totally bounded, if and only if its image under the canonical quotient map is totally bounded. Every relatively compact set is totally bounded and the closure of a totally bounded set is totally bounded. The image of a totally bounded set under a uniformly continuous map (such as a continuous linear map for instance) is totally bounded. If is a subset of a TVS such that every sequence in has a cluster point in then is totally bounded. If is a compact subset of a TVS and is an open subset of containing then there exists a neighborhood of 0 such that Closure and closed setThe closure of any convex (respectively, any balanced, any absorbing) subset of any TVS has this same property. In particular, the closure of any convex, balanced, and