Basis of a topology

A basis of a topology is a family of open sets such that every open set is a union of members of that family. Equivalently, every point of every open set lies in a basis member contained in it. The rectangles $U\times V$ form a basis for the product topology of two spaces. This topological meaning differs from a linear-algebra basis.

A family of open subsets is a subbasis of a topology when its finite intersections form a basis of a topology, including the empty intersection equal to the entire space. For the product topology, inverse images of open sets under the product projections form the standard subbasis of a topology. A map into the space is continuous if the inverse image of every subbasic set is open.