# Base or Open Base of a Topology

Let be a topological space, then the sub collection of is said to be a base or bases or open base for if each member of can be expressed as a union of members of .

In other words let be a topological space, then the sub collection of is said to be a base if for a point belonging to an open set there exists such that .

**Example:**

Let and let be a topology defined on . is a sub collection of , which meets the requirement for a base, because each member of is a union of members of .

**Remarks:**

• It should be noted that there may be more than one base for a given topology defined on that set.

• Since the union of an empty sub collection of members of is an empty set, so an empty set .

**For Discrete Topology**

Let and let be a topology defined on . is a base for . Check whether is a base or not,and take all possible unions of there must become .

Possible unions

In this case, the discrete topological space, the collection of all singletons subsets of , forms a base for a discrete topological space.

**For Indiscrete Topology**

Let and let be a topology defined on . is a base for .

**Theorem**

Let be a topological space, then a sub collection of is a base for if and only if:

1.

2. If and belongs to , then can be written as a union of members of ; i.e. for then there exist of such that .