# Subbase for a Topology

Let be a topological space. A sub-collection of subsets of is said to be an open subbase for or a subbase for topology if all finite intersections of members of form a base for .

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

**Example:**

Consider the Cartesian plane with usual topology. The is the base for the topological space , then the collection of all intervals of the form , where and gives a subbase for . Since the finite intersection of all such intervals gives the members of the base of , i.e., .

**Example:**

Let with topology

is a subbase for .

**Theorems**

• Let be a non-empty collection of subsets of . Suppose that , then is a subbase for some topology on .

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