# Base or Open Base of a Topology

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

In other words let be a topological space, then the sub collection of is said to be base, if for a point belonging to an open set then there exist 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 may be noted that there may be more than one base for a given topology defined on that set.

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

**For Discrete Topology:**

Let and let be a topology defined on . is a base for . Check weather is a base or not take all possible union of there must becomes .

Possible Unions

In this case discrete topological space, the collection of all singletons subsets of forms a base for 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 union of members of . i.e. for then there exist of such that .