up:: Metric Topology Let be a Metric Space, and an open set. Then, for each , there is some open ball such that . Thus, . Consequently, it is equal to , since it contains all of its points and not any more. Thus, any open set is a union of open balls. I.e. the set of all open balls in is a Topological Basis for ‘s Metric Topology. References Notas para um Curso de Física-Matemática, João Carlos Alves Barata. Cap. 24 Notas para um Curso de Física-Matemática, João Carlos Alves Barata. Cap. 27