# Basis for a Topology

Basis for a Topology

A basis (or base) for a topology on a set is a collection of open sets (the basis elements) such that every open set in is the union or finite intersection of members of .

X

B

X

B

Equivalently, a collection of open sets is a basis for a topology on if and only if it has the following properties:

B

X

1. For each , there is at least one basis element containing .

x∈X

B

x

2. If ,∈B and , then there is a basis element ∈B containing such that ⊂⋂.

B

1

B

2

x∈⋂

B

1

B

2

B

3

x

B

3

B

1

B

2

The set of all open disks contained in an open square form a basis. Drag the point within the square; then drag the centers of the disks and change their radii as needed to illustrate property 2 of a basis.