Let be a topological space and , then the sub collection is said to be local bases at a point if for belonging to an open set , there exists a member of , such that .
It can be defined as, let be a topological space and . A sub collection of is said to be a neighborhood base at a point or local base 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 base for a topology is also a local base at each point of ground set, but the converse may not be true.
• The union of all local bases forms bases for topology defined on 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 . This shows that is also a local base at point . Similarly, and .
Now which forms a base for .
It may be noted that the above procedure of finding local bases is only valid when the number of open sets containing a point is finite.
• Consider (Cartesian plane) with usual topology, and let be any point of , then the collection of all open discs with a center at form a local base at .
• Every discrete topological space has a countable neighborhood base at each of its points.