Subbase for a Topology

Let \left( {X,\tau } \right) be a topological space. A sub-collection S of subset of Xis said to be an open subbase forX or a subbase for topology \tau if all finite intersection of members of S forms 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 forms a base for topology \tau on X. The topology obtained in this way is called the topology generated by S.

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

Example:

Consider the Cartesian plane \mathbb{R} with usual topology the {\rm B} be 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 Xbe 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.

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