# Subbase for a Topology

Let $\left( {X,\tau } \right)$ be a topological space. A sub-collection $S$ of subsets of $X$ is said to be an open subbase for $X$ or a subbase for topology $\tau$ if all finite intersections of members of $S$ form a base for $\tau$.

In others words, a class $S$ of open sets of a space $X$ is called a subbase for a topology $\tau$ on $X$ if and only if intersections of members of $S$ form a base for topology $\tau$ on $X$. The topology obtained in this way is called the topology generated by $S$.

$\begin{array}{*{20}{c}} {\text{S}}&{\xrightarrow[{{\text{Finite intersections S}}}]{}}&{\rm B}&{\xrightarrow[{{\text{All union of members of }}{\rm B}}]{}}&\tau \end{array}$

Example:

Consider the Cartesian plane $\mathbb{R}$ with usual topology. The ${\rm B}$ is the base for the topological space $\mathbb{R}$, then the collection $S$ of all intervals of the form $\left] { – \infty ,b} \right[$, $\left] {a,\infty } \right[$ where $a,b \in \mathbb{R}$ and $a < b$ gives a subbase for $\mathbb{R}$. Since the finite intersection of all such intervals gives the members of the base of $\mathbb{R}$, i.e., $\left] { – \infty ,b} \right[ \cap \left] {a,\infty } \right[ = \left] {a,b} \right[$.

Example:

Let $X = \left\{ {a,b,c,d} \right\}$ with topology $\tau = \left\{ {\phi ,X,\left\{ a \right\},\left\{ c \right\},\left\{ d \right\},\left\{ {a,c} \right\},\left\{ {c,d} \right\},\left\{ {a,d} \right\},\left\{ {a,c,d} \right\}} \right\}$
${\rm B} = \left\{ {\left\{ a \right\},\left\{ c \right\},\left\{ d \right\},X} \right\}$
$S = \left\{ {\left\{ {a,c} \right\},\left\{ {c,d} \right\},\left\{ {a,d} \right\},X} \right\}$ is a subbase for $\tau$.

Theorems

• Let $S$ be a non-empty collection of subsets of $X$. Suppose that $X = \cup S$, then $S$ is a subbase for some topology on $X$.
• Let $X$ be any non-empty set, and let $S$ be an arbitrary collection of subsets of $X$. Then $S$ can serve as an open subbase for a topology on $X$, in the sense that the class of all unions of finite intersections of sets in $S$ is a topology.