In this book the authors introduce four types of topological vector subspaces. All topological vector subspaces are defined depending on a set. We define a quasi set topological vector subspace of a vector space depending on the subset S contained in the field F over which the vector space V is defined. These quasi set topological vector subspaces defined over a subset can be of finite or infinite dimension. An interesting feature about these spaces is that there can be several quasi set topological vector subspaces of a given vector space. This property helps one to construct several spaces with varying basic sets. Further we cannot define quasi set topological vector subspaces of all vector subspaces. We have given the number of quasi set topological vector subspaces in case of a vector space defined over a finite field. It is still an open problem, “Will these quasi set topological vector spaces increase the number of finite topological spaces with n points, n a finite positive integer?”
