A topological basis is a subset B of a set T in which all other open sets can be written as unions or finite intersections of B. For the real numbers, the set of all open intervals is a basis. Stated another way, if X is a set, a basis for a topology on X is a collection B of subsets of X (called basis elements) satisfying the following properties. 1. For each x element X, there is at least one basis element B containing x. 2. If x belongs to the intersection of two basis elements B_1 and B_2, then there is a basis element B_3 containing x such that B_3 subset B_1 intersection B_2. (Munkres 2000).