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 for $$x$$ belonging to an open set $$U$$, there exists 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 a neighborhood base at a point $$x$$ or local base 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 $$x \in B \subseteq U$$.

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

 

Example:
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 $$\left\{ a \right\} \cap \left\{ {a,b} \right\} \cap X = \left\{ a \right\}$$. This shows that $${{\rm B}_a} = \left\{ {\left\{ a \right\}} \right\}$$ is also a local base 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 base for $$\tau $$.

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.

Example:
• Consider $${\mathbb{R}^2}$$(Cartesian plane) with usual topology, and let $$x$$ be any point of $${\mathbb{R}^2}$$, then the collection of all open discs with a center at $$x$$ form a local base at $$x$$.
• Every discrete topological space has a countable neighborhood base at each of its points.