# 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 base or bases or open base for $\tau$, if each member of $\tau$ can be expressed as 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 base, if for a point $x$ belonging to an open set $U$ then there exist $B \in {\rm B}$ 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 may be noted that there may be more than one base for a given topology defined on that set.
• Since union of empty sub collection of members of ${\rm B}$ is an empty set, so 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 weather ${\rm B}$ is a base or not take all possible union of ${\rm B}$ there must becomes $\tau$.

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 discrete topological space, the collection of all singletons subsets of $X$ forms a base for 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 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 $x \in B \subseteq {B_1} \cap {B_2}$.