# Base or Open Base of a Topology

Let $$\left( {X,\tau } \right)$$ be a topological space, then the sub collection $${\rm B} $$ of $$\tau $$ is said to be a base or bases or open base for $$\tau $$ if each member of $$\tau $$ can be expressed as a union of members of $${\rm B}$$.

In other words let $$\left( {X,\tau } \right)$$ be a topological space, then the sub collection $${\rm B}$$ of $$\tau $$ is said to be a base if for a point $$x$$ belonging to an open set $$U$$ there exists $$B \in {\rm B}$$ such that $$x \in B \subseteq U$$.

**Example:**

Let $$X = \left\{ {a,b,c,d,e} \right\}$$ and let $$\tau = \left\{ {\phi ,\left\{ {a,b} \right\},\left\{ {c,d} \right\},\left\{ {a,b,c,d} \right\},X} \right\}$$ be a topology defined on $$X$$. $${\rm B} = \left\{ {\left\{ {a,b} \right\},\left\{ {c,d} \right\},X} \right\}$$ is a sub collection of $$\tau $$, which meets the requirement for a base, because each member of $$\tau $$ is a union of members of $${\rm B}$$.

**Remarks:**

• It should be noted that there may be more than one base for a given topology defined on that set.

• Since the union of an empty sub collection of members of $${\rm B}$$ is an empty set, so an empty set $$\phi \in \tau $$.

**For Discrete Topology**

Let $$X = \left\{ {1,2,3} \right\}$$ and let $$\tau = \left\{ {\phi ,\left\{ 1 \right\},\left\{ 2 \right\},\left\{ 3 \right\},\left\{ {1,2} \right\},\left\{ {2,3} \right\},\left\{ {1,3} \right\},X} \right\}$$ be a topology defined on $$X$$. $${\rm B} = \left\{ {\left\{ 1 \right\},\left\{ 2 \right\},\left\{ 3 \right\}} \right\}$$ is a base for $$\tau $$. Check whether $${\rm B}$$ is a base or not,and take all possible unions of $${\rm B}$$ there must become $$\tau $$.

Possible unions $$ = \left\{ {\phi ,\left\{ 1 \right\},\left\{ 2 \right\},\left\{ 3 \right\},\left\{ {1,2} \right\},\left\{ {2,3} \right\},\left\{ {1,3} \right\},X} \right\}$$

In this case, the discrete topological space, the collection of all singletons subsets of $$X$$, forms a base for a discrete topological space.

**For Indiscrete Topology**

Let $$X = \left\{ {1,2,3} \right\}$$ and let $$\tau = \left\{ {\phi ,X} \right\}$$ be a topology defined on $$X$$. $${\rm B} = \left\{ X \right\}$$ is a base for $$\tau $$.

**Theorem**

Let $$\left( {X,\tau } \right)$$ be a topological space, then a sub collection $${\rm B}$$of $$\tau $$ is a base for $$\tau $$ if and only if:

1. $$X = \bigcup\limits_{B \in {\rm B}} B $$

2. If $${B_1}$$ and $${B_2}$$ belongs to $${\rm B}$$, then $${B_1} \cap {B_2}$$ can be written as a union of members of $${\rm B}$$; i.e. for $$x \in {B_1} \cap {B_2}$$ then there exist $$B$$ of $${\rm B}$$ such that $$x \in B \subseteq {B_1} \cap {B_2}$$.