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 .
• 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 .
Let be a non-empty set with topology defined on . Consider the open sets containing are then is a local base at point .
Since , and .
Note that . Which shows that is also a local bases at point . Similarly, and .
Now which forms a bases for .
It may note that above procedure of finding local bases is only valid when a number of open sets containing a point is finite.
• Consider (Cartesian plane) with usual topology, and let be any point of , then collection of all open discs with centre at form a local bases at .
• Every discrete topological space has a countable neighbourhood base at each of its points.