Let  be a topological space and , then the sub collection  is said to be local bases at a point , if  belonging to an open set , there exist a member  of , such that .
            It can be defined as, let  be a topological space and. A sub collection  of is said to be neighbourhood base at a point  or local bases at a point  or simply a base at a point , if for any open set  containing , there is a  such that .

Remark:

• It may be noted that every bases for a topology is also a local base at each point of ground set but the converse may not be true.
• Union of all local bases forms bases for topology  defined on the any non-empty set X.

Example:
            Let  be a non-empty set with topology  defined on X. Consider the open sets containing “a” are  then is a local base at point “a”.
            Since ,  and .
            Note that . Which shows that  is also a local bases at point “a”. Similarly,  and .
            Now  which forms a bases for .
It may noted that above procedure of finding local bases is only valid when a number of open sets containing a point is finite.

Example:

• Consider (Cartesian plane) with usual topology, and let x be any point of , then collection of all open discs with centre at x form a local bases at x.
• Every discrete topological space has a countable neighbourhood base at each of its points.