Let be a topological space, then the sub collection of  is said to be base or bases or open base for, if each member of  can be expressed as union of members of .
            In other words let be a topological space, then the sub collection  of  is said to be base, if for a point belonging to an open set U then there exist such that .

Example:
            Let  and let  be a topology defined on X.  is a sub collection of , which meets the requirement for a base, because each member of  is a union of members of .

Remarks:

• It may be noted that there may be more than one base for a given topology defined on that set.
• Since union of empty sub collection of members of  is an empty set, so empty set .

For Discrete Topology:
                Let  and let  be a topology defined on X.  is a base for. Check weather is a base or not take all possible union of  there must becomes.
            Possible Unions
In this case discrete topological space, the collection of all singletons subsets of X forms a base for discrete topological space.

For Indiscrete Topology:
             Let  and let  be a topology defined on X.  is a base for.

Theorem:
            Let be a topological space, then a sub collection of  is a base for  if and only if

• 
• If  and belongs to, then can be written as union of members of . i.e. for then there exist of  such that .