Let  be a topological space. A sub-collection of subset of is said to be an open subbase for or a subbase for topology if all finite intersection of members of  forms a base for .
            In others words, A class of open sets of a space  is called a subbase for a topology  on , if and only if intersections of members of  forms a base for topology  on . The topology obtained in this way is called the topology generated by .

Example:
            Consider the Cartesian plane  with usual topology the  be the base for the topological space , then the collection  of all intervals of the form ,  where  and  gives a subbase for . Since the finite intersection of all such intervals gives the members of the base of , i.e., .

Example:
            Let  with topology
           
             is a subbase for .

Theorems:

• Let  be a non-empty collection of subsets of . Suppose that , then  is a subbase for some topology on .
• Let be any non-empty set, and let  be an arbitrary collection of subsets of . Then  can serve as an open subbase for a topology on , in the sense that the class of all unions of finite intersections of sets in  is a topology.