# 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.