A topological basis is a subset of a set in which all other open sets can be written as unions or finite intersections of . For the real numbers, the set of all open intervals is a basis.

Stated another way, if is a set, a basis for a topology on is a collection of subsets of (called basis elements) satisfying the following properties.

1. For each , there is at least one basis element containing .

2. If belongs to the intersection of two basis elements and , then there is a basis element containing such that .

(Munkres 2000).