Topological Basis

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 .

REFERENCES:

Munkres, J. R. Topology: A First Course, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 2000.