The concept of ε-enlargement defined on metric spaces is generalized to the concept of U-enlargement by using neighborhoods U of the zero of the space on topological vector spaces. By using U-enlargement, we define the bornological convergence for nets of sets in topological vector spaces and we examine some of their properties. By using filters defined on natural numbers, we define the concept of filter bornological convergence on sequences of sets, which is a more general concept than the bornological convergence defined on topological vector spaces. We give similar results for the filter bornological convergence