neighborhood basis at the origin for any vector topology. Defining topologies using strings Let be a vector space and let be a sequence of subsets of Each set in the sequence is called a of and for every index is called the -th knot of The set is called the beginning of The sequence is/is a: if for every index Balanced (resp. absorbing, closed, convex, open, symmetric, barrelled, absolutely convex/disked, etc.) if this is true of every if is summative, absorbing, and balanced. or a in a TVS if is a string and each of its knots is a neighborhood of the origin in If is an absorbing disk in a vector space then the sequence defined by forms a string beginning with This is called the natural string of Moreover, if a vector space has countable dimension then every string contains an absolutely convex string. Summative sequences of sets have the particularly nice property that they define non-negative continuous real-valued subadditive functions. These functions can then be used to prove many of the basic properties of topological vector spaces. A proof of the above theorem is given in the article on metrizable topological vector spaces. If and are two collections of subsets of a vector space and if is a scalar, then by definition: contains : if and only if for every index Set of knots: Kernel: Scalar multiple: Sum: Intersection: If is a collection sequences of subsets of then is said to be directed (downwards) under inclusion or simply directed downward if is not empty and for all there exists some such that and (said differently, if and only if is a prefilter with respect to the containment defined above). Notation: Let be the set of all knots of all strings in Defining vector topologies using collections of strings is particularly useful for defining classes of TVSs that are not necessarily locally convex. If is the set of all topological strings in a TVS then A Hausdorff TVS is metrizable if and only if its topology can be induced by a single topological string. Topological structure A vector space is an abelian group with respect to the operation of addition, and in a topological vector space the inverse operation is always continuous (since it is the same as multiplication by ). Hence, every topological vector space is an abelian topological group. Every TVS is completely regular but a