conditions for a collection of subsets to be a basis for some topology

conditions for a collection of subsets to be a basis for some topology

Not just any

collection

of subsets of

X

can be a basis for a

topology

on

X

. For

instance

, if we took

𝒞

to be all

open intervals

of length

1

in

ℝ

,

𝒞

isn’t the basis for any topology on

ℝ

:

(

0

,

1

)

and

(

.5

,

1.5

)

are unions of elements of

𝒞

, but their

intersection

(

.5

,

1

)

is not. The collection formed by arbitrary unions of members of

𝒞

isn’t

closed under

finite intersections and isn’t a topology.

We’d like to know which collections

ℬ

of subsets of

X

could be the basis for some topology on

X

. Here’s the result:

Theorem

.

A collection

ℬ

of subsets of

X

is a basis for some topology on

X

if and only if:

1.

Every

x

∈

X

is

contained

in some

B

x

∈

ℬ

, and

2.

If

B

1

and

B

2

are two elements of

ℬ

containing

x

∈

X

, then there’s a third

element

B

3

of

ℬ

such that

x

∈

B

3

⊂

B

1

∩

B

2

.

Proof.

First, we’ll show that if

ℬ

is the basis for some topology

𝒯

on

X

, then it

satisfies

the two conditions listed.

𝒯

is a topology on

X

, so

X

∈

𝒯

. Since

ℬ

is a basis for

𝒯

, that means

X

can be written as a union of members of

ℬ

: since every

x

∈

X

is in this union, every

x

∈

X

is contained in some member of

ℬ

. That takes care of the first condition.

For the second condition: if

B

1

and

B

2

are elements of

ℬ

, they’re also in

𝒯

.

𝒯

is closed under intersection, so

B

1

∩

B

2

is open in

𝒯

. Then

B

1

∩

B

2

can be written as a union of members of

ℬ

, and any

x

∈

B

1

∩

B

2

is contained by some basis element in this union.

Second, we’ll show that if a collection

ℬ

of subsets of

X

satisfies the two conditions, then the collection

𝒯

of unions of members of

ℬ

is a topology on

X

.

•

∅

∈

𝒯

:

∅

is the null union of zero elements of

ℬ

.

•

X

∈

𝒯

: by the first condition, every

X

is contained in some member of

ℬ

. The union of all the members of

ℬ

is then all of

X

.

•

𝒯

is closed under arbitrary unions: Say we have a union of sets

T

α

∈

𝒯

…

⋃

α

∈

I

T

α

=

⋃

α

∈

I

⋃

β

∈

J

α

B

β

(since each

T

α

is a union of sets in

ℬ

)

=

⋃

β

∈

⋃

α

∈

I

J

α

B

β

Since that’s a union of elements of

ℬ

, it’s also a member of

𝒯

.

•

𝒯

is closed under finite intersections: since a collection of sets is closed under finite intersections if and only if it is closed under pairwise intersections, we need only check that the intersection of two members

T

1

,

T

2

of

𝒯

is in

𝒯

.

Any

x

∈

T

1

∩

T

2

is contained in some

B

x

1

⊂

T

1

and

B

x

2

⊂

T

2

. By the second condition,

x

∈

B

x

1

∩

B

x

2

gets us a

B

x

3

with

x

∈

B

x

3

⊂

B

x

1

∩

B

x

2

⊂

T

1

∩

T

2

. Then

T

1

∩

T

2

=

⋃

x

∈

T

1

∩

T

2

B

x

3

which is in

𝒯

.

∎

Title

conditions for a collection of subsets to be a basis for some topology

Canonical name

ConditionsForACollectionOfSubsetsToBeABasisForSomeTopology

Date of creation

2013-03-22 14:21:49

Last modified on

2013-03-22 14:21:49

Owner

waj (4416)

Last modified by

waj (4416)

Numerical id

4

Author

waj (4416)

Entry type

Proof

Classification

msc 54A99

Classification

msc 54D70

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