# 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.