Basis for a Topology

radius of

B

1

radius of

B

2

radius of

B

3

B 1 ⋂ B 2 ≠ ∅	False
x ∈ B 1	False
x ∈ B 2	False
B 3 ⊂ B 1 ⋂ B 2	False
x ∈ B 3	False

A basis (or base) for a topology on a set

X

is a collection of open sets

B

(the basis elements) such that every open set in

X

is the union or finite intersection of members of

B

.

Equivalently, a collection of open sets

B

is a basis for a topology on

X

if and only if it has the following properties:

1. For each

x∈X

, there is at least one basis element

B

containing

x

.

2. If

B

1

,

B

2

∈B

and

x∈

B

1

⋂

B

2

, then there is a basis element

B

3

∈B

containing

x

such that

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.