Local Base for a Topology

Let \left( {X,\tau } \right) be a topological space and x \in X, then the sub collection {{\rm B}_x} is said to be local bases at a point x, if x belonging to an open set U, there exist a member B of {\rm B}, such that x \in B \subseteq U.

It can be defined as, let \left( {X,\tau } \right) be a topological space and x \in X. A sub collection {\rm B} of \tau is said to be neighbourhood base at a point x or local bases at a point x or simply a base at a point x, if for any open set U containing x, there is a B \in {\rm B} such that x \in B \subseteq U.

• 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 \tau defined on the any non-empty set X.

Let X = \left\{ {a,b,c} \right\} be a non-empty set with topology \tau  = \left\{ {\phi ,X,\left\{ a \right\},\left\{ b \right\},\left\{ {a,b} \right\},\left\{ {b,c} \right\}} \right\} defined on X. Consider the open sets containing a are X,\left\{ a \right\},\left\{ {a,b} \right\} then {{\rm B}_a} = \left\{ {X,\left\{ a \right\},\left\{ {a,b} \right\}} \right\}is a local base at point a.

Since a \in \left\{ a \right\} \subseteq \left\{ a \right\}, a \in \left\{ {a,b} \right\} \subseteq \left\{ {a,b} \right\} and a \in X \subseteq X.

Note that \left\{ a \right\} \cap \left\{ {a,b} \right\} \cap X = \left\{ a \right\}. Which shows that {{\rm B}_a} = \left\{ {\left\{ a \right\}} \right\} is also a local bases at point a. Similarly, {{\rm B}_b} = \left\{ {\left\{ b \right\}} \right\} and {{\rm B}_c} = \left\{ {\left\{ {b,c} \right\}} \right\}.
Now {\rm B} = {{\rm B}_a} \cup {{\rm B}_b} \cup {{\rm B}_c} = \left\{ {\left\{ a \right\},\left\{ b \right\},\left\{ {b,c} \right\}} \right\} which forms a bases for \tau .

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 {\mathbb{R}^2}(Cartesian plane) with usual topology, and let x be any point of {\mathbb{R}^2}, 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.