aarjaneiro/planetmath huggingface dataset:
proof of properties of the closure operator

proof of properties of the closure operator

Recall that the

closure

of a set

A

in a

topological space

X

is defined to be the

intersection

of all

closed sets

containing it.

A

⊂

A

¯

: By

definition

A

¯

=

⋂

C

⊇

A

,

C

⁢

closed

C

,

but since for every

C

we have

A

⊆

C

, we immediately find

A

⊆

⋂

C

⊇

A

,

C

⁢

closed

C

.

A

¯

is closed

: Recall that the intersection of any number of closed sets is closed, so the closure is itself closed.

∅

¯

=

∅

,

X

¯

=

X

, and

A

¯

¯

=

A

¯

:
If

C

is any closed set, then

C

¯

=

⋂

C

′

⊇

C

,

C

′

⁢

closed

C

′

=

C

∩

⋂

C

′

⊋

C

,

C

′

⁢

closed

C

′

=

C

.

A

∪

B

¯

=

A

¯

∪

B

¯

: First write down the definition:

A

¯

∪

B

¯

=

⋂

C

⊇

A

,

C

⁢

closed

C

∪

⋂

D

⊇

B

,

D

⁢

closed

D

,

then apply DeMorgan’s law to get

=

⋂

C

⊇

A

,

D

⊇

B

,

C

,

D

⁢

closed

(

C

∪

D

)

,

but for every such pair

C

,

D

, we have that

E

=

C

∪

D

is a closed set containing

A

∪

B

.

Conversely

, every closed set

E

containing

A

∪

B

is obtained from such a pair — just take

(

E

,

E

)

to be the pair. Thus

=

⋂

E

⊇

A

∪

B

,

E

⁢

closed

(

E

)

=

A

∪

B

¯

.

A

∩

B

¯

⊂

A

¯

∩

B

¯

:

A

¯

∩

B

¯

=

⋂

C

⊇

A

,

C

⁢

closed

C

∩

⋂

D

⊇

B

,

D

⁢

closed

D

,

=

⋂

C

⊇

A

,

D

⊇

B

,

C

,

D

⁢

closed

(

C

∩

D

)

,

but for every such pair

C

,

D

, we have that

E

=

C

∩

D

is a closed set containing

A

∩

B

. However, some closed sets may not arise in this way, so we do not have equality. Thus

⊇

⋂

E

⊇

A

∩

B

,

E

⁢

closed

(

E

)

=

A

∩

B

¯

.

so we have

A

¯

∩

B

¯

⊇

A

∩

B

¯

.

A

¯

=

A

∪

A

′

where

A

′

is the set of all

limit points

of

A

:
Let

a

be a limit point of

A

, and let

C

be a closed set containing

A

. If

a

is not in

C

, then

X

∖

C

is an

open set

containing

a

but not meeting

C

, which implies that

X

∖

C

does not meet

A

, which contradicts the fact that

a

was a limit point of

A

. Conversely, suppose that

a

is not a limit point of

A

, and that

a

is not in

A

. Then there is some open neighborhood

U

of

a

which does not meet

A

. But then

X

∖

U

is a closed set containing

A

but not containing

a

, so

a

∉

A

¯

.

Title

proof of

properties

of the

closure operator

Canonical

name

ProofOfPropertiesOfTheClosureOperator

Date of creation

2013-03-22 14:12:19

Last modified on

2013-03-22 14:12:19

Owner

archibal (4430)

Last modified by

archibal (4430)

Numerical id

4

Author

archibal (4430)

Entry type

Proof

Classification

msc 54A99

Generated on Sat Feb 10 11:16:15 2018 by

LaTeXML
proof that components of open sets in a locally connected space are open

proof that components of open sets in a locally connected space are open

Theorem

.

A

topological space

X

is

locally connected

if and only if each component of an

open set

is open.

Proof.

First, suppose that

X

is locally connected and that

U

is an open set of

X

.
Let

p

∈

C

, where

C

is a component of

U

.
Since

X

is locally connected there is an open connected set, say

V

with

p

∈

V

⊂

U

. Since

C

is a component of

U

it must be that

V

⊂

C

.
Hence,

C

is open.
For the converse, suppose that each component of each open set is open. Let

p

∈

X

.
Let

U

be an open set containing

p

. Let

C

be the component of

U

which
contains

p

. Then

C

is open and

connected

, so

X

is locally connected.

∎

As a corollary, we have that the components of a locally connected space are both
open and closed.

Title

proof that components of open sets in a locally connected space are open

Canonical name

ProofThatComponentsOfOpenSetsInALocallyConnectedSpaceAreOpen

Date of creation

2013-03-22 17:06:07

Last modified on

2013-03-22 17:06:07

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

6

Author

Mathprof (13753)

Entry type

Theorem

Classification

msc 54A99

Generated on Sat Feb 10 11:16:17 2018 by

LaTeXML
properties of the closure operator

properties of the closure operator

Suppose

X

is a

topological space

, and let

A

¯

be the

closure

of

A

in

X

.
Then the following properties hold:

1.

A

¯

=

A

∪

A

′

where

A

′

is the

derived set

of

A

.

2.

A

⊆

A

¯

, and

A

=

A

¯

if and only if

A

is closed

3.

A

¯

=

∅

if and only if

A

=

∅

.

4.

If

Y

is another topological space, then

f

:

X

→

Y

is a

continuous map

,
if and only if

f

⁢

(

A

¯

)

⊆

f

⁢

(

A

)

¯

for all

A

⊆

X

. If

f

is also a

homeomorphism

,
then

f

⁢

(

A

¯

)

=

f

⁢

(

A

)

¯

.

Title

properties of the

closure operator

Canonical name

PropertiesOfTheClosureOperator

Date of creation

2013-03-22 15:17:05

Last modified on

2013-03-22 15:17:05

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

11

Author

matte (1858)

Entry type

Theorem

Classification

msc 54A99

Generated on Sat Feb 10 11:16:20 2018 by

LaTeXML
section filter

section filter

Let

X

be a set and

(

x

i

)

i

∈

D

a non-empty net in

X

. For each

j

∈

D

, define

S

⁢

(

j

)

:=

{

x

i

∣

i

≤

j

}

. Then the set

S

:=

{

S

⁢

(

j

)

∣

j

∈

D

}

is a

filter basis

:

S

is non-empty because

(

x

i

)

≠

∅

, and for any

j

,

k

∈

D

, there is a

ℓ

such that

j

≤

ℓ

and

k

≤

ℓ

, so that

S

⁢

(

ℓ

)

⊆

S

⁢

(

j

)

∩

S

⁢

(

k

)

.

Let

𝒜

be the family of all filters containing

S

.

𝒜

is non-empty since the

filter generated by

S

is in

𝒜

. Order

𝒜

by

inclusion

so that

𝒜

is a poset. Any chain

ℱ

1

⊆

ℱ

2

⊆

⋯

has an

upper bound

, namely,

ℱ

:=

⋃

i

=

1

∞

ℱ

i

.

By Zorn’s lemma,

𝒜

has a

maximal element

𝒳

.

Definition

.

𝒳

defined above is called the

section filter

of the net

(

x

i

)

in

X

.

Remark

. A section filter is obviously a filter. The name “

section

” comes from the

elements

S

⁢

(

j

)

of

S

, which are sometimes known as “sections” of the net

(

x

i

)

.

Title

section filter

Canonical

name

SectionFilter

Date of creation

2013-03-22 16:41:37

Last modified on

2013-03-22 16:41:37

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

4

Author

CWoo (3771)

Entry type

Definition

Classification

msc 54A99

Classification

msc 03E99

Generated on Sat Feb 10 11:16:30 2018 by

LaTeXML
star refinement

star refinement

Let

X

be a set and

𝒞

=

{

C

i

∣

i

∈

I

}

be a cover of

X

(we assume

C

i

and

X

are all subsets of some universe). Let

A

⊆

X

. The

star

of

A

(with respect to the cover

𝒞

) is defined as

⋆

(

A

,

𝒞

)

:=

⋃

{

C

i

∈

𝒞

∣

C

i

∩

A

≠

∅

}

.

When

A

is a

singleton

, we write

⋆

(

x

,

𝒞

)

=

⋆

(

{

x

}

,

𝒞

)

.

Properties

of

⋆

1.

A

⊆

⋆

(

A

,

𝒞

)

.

2.

If

A

⊆

B

, then

⋆

(

A

,

𝒞

)

⊆

⋆

(

B

,

𝒞

)

.

3.

For any cover

𝒞

of

X

, the sets

𝒞

⋆

:=

{

⋆

(

C

i

,

𝒞

)

∣

C

i

∈

𝒞

}

and

𝒞

b

:=

{

⋆

(

x

,

𝒞

)

∣

x

∈

X

}

are both covers of

X

.

4.

𝒞

⪯

𝒞

b

⪯

𝒞

⋆

(

⪯

denotes

cover refinement

).

Definitions

. Let

𝒞

,

𝒟

be two covers of

X

. If

𝒞

⋆

⪯

𝒟

, then we say that

𝒞

is a

star refinement

of

𝒟

, denoted by

𝒞

⪯

⋆

𝒟

. If

𝒞

b

⪯

𝒟

, then we say that

𝒞

is a

barycentric refinement

of

𝒟

, denoted by

𝒞

⪯

b

𝒟

.

Remark

. By property 4 above, it is easy to see that

𝒞

⪯

⋆

𝒟

⇒

𝒞

⪯

b

𝒟

⇒

𝒞

⪯

𝒟

.

References

1

S. Willard,

General Topology

,
Addison-Wesley, Publishing Company, 1970.

Title

star refinement

Canonical

name

StarRefinement

Date of creation

2013-03-22 16:44:13

Last modified on

2013-03-22 16:44:13

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 54A99

Defines

star

Defines

star refine

Defines

barycentric refinement

Generated on Sat Feb 10 11:16:40 2018 by

LaTeXML
subbasis

subbasis

Let

(

X

,

𝒯

)

be a

topological space

. A subset

𝒜

⊆

𝒯

is said to be a

subbasis

if the

collection

ℬ

of

intersections

of finitely many

elements of

𝒜

is a basis (

http://planetmath.org/BasisTopologicalSpace

) for

𝒯

.

Conversely

, given an arbitrary collection

𝒜

of subsets of

X

, a

topology

can be formed by first taking the collection

ℬ

of finite intersections of members of

𝒜

and then taking the topology

𝒯

generated by

ℬ

as basis.

𝒯

will then be the smallest topology such that

𝒜

⊆

𝒯

.

Title

subbasis

Canonical name

Subbasis

Date of creation

2013-03-22 12:10:32

Last modified on

2013-03-22 12:10:32

Owner

evin290 (5830)

Last modified by

evin290 (5830)

Numerical id

10

Author

evin290 (5830)

Entry type

Definition

Classification

msc 54A99

Synonym

subbasic

Synonym

subbasic

Synonym

subbase

Related topic

Basis

Related topic

BasisTopologicalSpace

Generated on Sat Feb 10 11:16:41 2018 by

LaTeXML
the union of a locally finite collection of closed sets is closed

the union of a locally finite collection of closed sets is closed

The union of a collection of

closed subsets

of a

topological space

need not,
of course, be closed. However, we do have the following result:

Theorem

.

The union of a

locally finite collection

of closed subsets
of a topological space is itself closed.

Proof.

Let

𝒮

be a locally finite collection of closed subsets
of a topological space

X

, and put

Y

=

∪

𝒮

.
Let

x

∈

X

∖

Y

.
By local finiteness there is an open

neighbourhood

U

of

x

that meets only finitely many members of

𝒮

,
say

A

1

,

…

,

A

n

.
So

U

∖

Y

=

U

∖

⋃

i

=

1

n

A

i

, which is open.
Thus

U

∖

Y

is an open neighbourhood of

x

that does not meet

Y

.
It follows that

Y

is closed.
∎

One use for this result can be found in the entry on

gluing together continuous functions

.

Title

the union of a locally finite collection of closed sets is closed

Canonical name

TheUnionOfALocallyFiniteCollectionOfClosedSetsIsClosed

Date of creation

2013-03-22 16:14:45

Last modified on

2013-03-22 16:14:45

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

10

Author

yark (2760)

Entry type

Theorem

Classification

msc 54A99

Generated on Sat Feb 10 11:16:43 2018 by

LaTeXML
topological sum

topological sum

Given two

topological spaces

X

and

Y

, their

topological sum

is defined
to be the set

X

⁢

∐

Y

(see the entry

disjoint union

) equipped with
the finest

topology

such that the

inclusion maps

from

X

and

Y

into

X

⁢

∐

Y

are

continuous

. A basis for this topology consists of the union of the set of

open subsets

of

X

and the set of open subsets of

Y

.

Title

topological sum

Canonical name

TopologicalSum

Date of creation

2013-03-22 14:41:29

Last modified on

2013-03-22 14:41:29

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

7

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 54A99

Synonym

coproduct

in the category of topological spaces

Synonym

topological disjoint union

Generated on Sat Feb 10 11:16:44 2018 by

LaTeXML
types of limit points

types of limit points

Let

X

be a

topological space

and

A

⊂

X

be a subset.

A point

x

∈

X

is an

ω

-

accumulation point

of

A

if every

open set

in

X

that contains

x

also contains infinitely many points of

A

.

A point

x

∈

X

is a

condensation point

of

A

if every open set in

X

that contains

x

also contains uncountably many points of

A

.

If

X

is in

addition

a

metric space

, then a

cluster point

of a sequence

{

x

n

}

is a point

x

∈

X

such that every

ϵ

>

0

, there are infinitely many point

x

n

such that

d

⁢

(

x

,

x

n

)

<

ϵ

.

These are all clearly examples of

limit points

.

Title

types of limit points

Canonical

name

TypesOfLimitPoints

Date of creation

2013-03-22 14:37:50

Last modified on

2013-03-22 14:37:50

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 54A99

Defines

ω

-accumulation points

Defines

condensation points

Defines

cluster points

Generated on Sat Feb 10 11:16:46 2018 by

LaTeXML
characterization of subspace topology

characterization of subspace topology

Theorem

.

Let

X

be a

topological space

and

Y

⊂

X

any subset. The

subspace topology

on

Y

is the weakest

topology

making the

inclusion map

continuous

.

Proof.

Let

𝒮

denote the subspace topology on

Y

and

j

:

Y

↪

X

denote the inclusion map.

Suppose

{

𝒯

α

∣

α

∈

J

}

is a family of topologies on

Y

such that each inclusion map

j

α

:

(

Y

,

𝒯

α

)

↪

X

is continuous. Let

𝒯

be the

intersection

⋂

α

∈

J

𝒯

α

. Observe that

𝒯

is also a topology on

Y

. Let

U

be open in

X

. By

continuity

of

j

α

, the set

j

α

-

1

⁢

(

U

)

=

j

-

1

⁢

(

U

)

is open in each

𝒯

α

; consequently,

j

-

1

⁢

(

U

)

is also in

𝒯

. This shows that there is a weakest topology on

Y

making inclusion continuous.

We claim that any topology strictly weaker than

𝒮

fails to make the inclusion map continuous. To see this, suppose

𝒮

0

⊊

𝒮

is a topology on

Y

. Let

V

be a set open in

𝒮

but not in

𝒮

0

. By the definition of subspace topology,

V

=

U

∩

Y

for some

open set

U

in

X

. But

j

-

1

⁢

(

U

)

=

V

, which was specifically chosen not to be in

𝒮

0

. Hence

𝒮

0

does not make the inclusion map continuous. This

completes

the proof.
∎

Title

characterization of subspace topology

Canonical name

CharacterizationOfSubspaceTopology

Date of creation

2013-03-22 15:40:32

Last modified on

2013-03-22 15:40:32

Owner

mps (409)

Last modified by

mps (409)

Numerical id

6

Author

mps (409)

Entry type

Theorem

Classification

msc 54B05

Generated on Sat Feb 10 11:16:48 2018 by

LaTeXML
closed set in a subspace

closed set in a subspace

In the following, let

X

be a

topological space

.

Theorem 1

.

Suppose

Y

⊆

X

is equipped with the

subspace topology

,
and

A

⊆

Y

.
Then

A

is closed (

http://planetmath.org/ClosedSet

) in

Y

if and only if

A

=

Y

∩

J

for some

closed set

J

⊆

X

.

Proof.

If

A

is closed in

Y

,
then

Y

∖

A

is open (

http://planetmath.org/OpenSet

) in

Y

,
and by the definition of the subspace topology,

Y

∖

A

=

Y

∩

U

for some open

U

⊆

X

.
Using properties of the

set difference

(

http://planetmath.org/SetDifference

),
we obtain

A

=

Y

∖

(

Y

∖

A

)

=

Y

∖

(

Y

∩

U

)

=

Y

∖

U

=

Y

∩

U

∁

.

On the other hand, if

A

=

Y

∩

J

for some closed

J

⊆

X

,
then

Y

∖

A

=

Y

∖

(

Y

∩

J

)

=

Y

∩

J

∁

,
and so

Y

∖

A

is open in

Y

,
and therefore

A

is closed in

Y

.
∎

Theorem 2

.

Suppose

X

is a topological space,

C

⊆

X

is a closed set equipped with the subspace topology,
and

A

⊆

C

is closed in

C

.
Then

A

is closed in

X

.

Proof.

This follows from the previous

theorem

:
since

A

is closed in

C

,
we have

A

=

C

∩

J

for some closed set

J

⊆

X

,
and

A

is closed in

X

.
∎

Title

closed set in a

subspace

Canonical name

ClosedSetInASubspace

Date of creation

2013-03-22 15:33:32

Last modified on

2013-03-22 15:33:32

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

9

Author

yark (2760)

Entry type

Theorem

Classification

msc 54B05

Generated on Sat Feb 10 11:16:50 2018 by

LaTeXML
irreducible component

irreducible component

An

irreducible component

of a

topological space

is a maximal

irreducible

subset.

If a subset is irreducible, its

closure

is, so irreducible components are closed.

Title

irreducible component

Canonical

name

IrreducibleComponent

Date of creation

2013-03-22 12:47:29

Last modified on

2013-03-22 12:47:29

Owner

nerdy2 (62)

Last modified by

nerdy2 (62)

Numerical id

6

Author

nerdy2 (62)

Entry type

Definition

Classification

msc 54B05

Classification

msc 14A10

Classification

msc 14A15

Generated on Sat Feb 10 11:16:52 2018 by

LaTeXML
subspace of a subspace

subspace of a subspace

Theorem 1

.

Suppose

X

⊆

Y

⊆

Z

are sets and

Z

is a

topological space

with

topology

τ

Z

.
Let

τ

Y

,

Z

be the

subspace topology

in

Y

given by

τ

Z

,
and let

τ

X

,

Y

,

Z

be the subspace topology in

X

given by

τ

Y

,

Z

, and let

τ

X

,

Z

be the subspace topology in

X

given by

τ

Z

. Then

τ

X

,

Z

=

τ

X

,

Y

,

Z

.

Proof.

Let

U

X

∈

τ

X

,

Z

, then there is by the definition of the subspace
topology an

open set

U

Z

∈

τ

Z

such that

U

X

=

U

Z

∩

X

. Now

U

Z

∩

Y

∈

τ

Y

,

Z

and therefore

U

Z

∩

Y

∩

X

∈

τ

X

,

Y

,

Z

. But since

X

⊆

Y

, we have

U

Z

∩

Y

∩

X

=

U

Z

∩

X

=

U

X

, so

U

X

∈

τ

X

,

Y

,

Z

and thus

τ

X

,

Z

⊆

τ

X

,

Y

,

Z

.

To show the reverse inclusion, take an open set

U

X

∈

τ

X

,

Y

,

Z

. Then there is an open set

U

Y

∈

τ

Y

,

Z

such that

U

X

=

U

Y

∩

X

. Furthermore, there is an open set

U

Z

∈

τ

Z

such that

U

Y

=

U

Z

∩

Y

. Since

X

⊆

Y

, we
have

U

Z

∩

X

=

U

Z

∩

Y

∩

X

=

U

Y

∩

X

=

U

X

,

so

U

X

∈

τ

X

,

Z

and thus

τ

X

,

Y

,

Z

⊆

τ

X

,

Z

.

Together, both inclusions yield the equality

τ

X

,

Z

=

τ

X

,

Y

,

Z

.
∎

Title

subspace

of a subspace

Canonical name

SubspaceOfASubspace

Date of creation

2013-03-22 15:17:53

Last modified on

2013-03-22 15:17:53

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

5

Author

matte (1858)

Entry type

Theorem

Classification

msc 54B05

Generated on Sat Feb 10 11:16:54 2018 by

LaTeXML
subspace topology

subspace topology

Let

X

be a

topological space

, and let

Y

⊂

X

be a subset. The

subspace topology

on

Y

is the

topology

whose

open sets

are those subsets of

Y

which equal

U

∩

Y

for some open set

U

⊂

X

.

In this context, the topological space

Y

obtained by taking the subspace topology is called a

topological subspace

, or simply

subspace

, of

X

.

Title

subspace topology

Canonical name

SubspaceTopology

Date of creation

2013-03-22 11:53:22

Last modified on

2013-03-22 11:53:22

Owner

djao (24)

Last modified by

djao (24)

Numerical id

8

Author

djao (24)

Entry type

Definition

Classification

msc 54B05

Classification

msc 15A66

Classification

msc 11E88

Synonym

relative topology

Defines

topological subspace

Defines

subspace

Generated on Sat Feb 10 11:16:56 2018 by

LaTeXML
subspace topology in a metric space

subspace topology in a metric space

Theorem 1

.

Suppose

X

is a

topological space

whose

topology

is induced by a
metric

d

, and suppose

Y

⊆

X

is a subset.
Then the

subspace topology

in

Y

is the same as the

metric topology

when by

d

restricted to

Y

.

Let

d

′

:

Y

:

Y

→

ℝ

be the

restriction

of

d

to

Y

,
and let

B

r

⁢

(

x

)

=

{

z

∈

X

:

d

′

(

z

,

x

)

<

r

}

,

B

r

′

⁢

(

x

)

=

{

z

∈

Y

:

d

′

(

z

,

x

)

<

r

}

.

The proof rests on the

identity

B

r

′

⁢

(

x

)

=

Y

∩

B

r

⁢

(

x

)

,

x

∈

Y

,

r

>

0

.

Suppose

A

⊆

Y

is open in the subspace topology of

Y

,
then

A

=

Y

∩

V

for some open

V

⊆

X

. Since

V

is open in

X

,

V

=

∪

{

B

r

i

⁢

(

x

i

)

:

i

=

1

,

2

,

…

}

for some

r

i

>

0

,

x

i

∈

X

, and

A

=

∪

{

Y

∩

B

r

i

⁢

(

x

i

)

:

i

=

1

,

2

,

…

}

=

∪

{

B

r

i

′

⁢

(

x

i

)

:

i

=

1

,

2

,

…

}

.

Thus

A

is open also in the metric topology of

d

′

.
The converse direction is proven similarly.

Title

subspace topology in a metric space

Canonical name

SubspaceTopologyInAMetricSpace

Date of creation

2013-03-22 15:17:44

Last modified on

2013-03-22 15:17:44

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

5

Author

matte (1858)

Entry type

Theorem

Classification

msc 54B05

Generated on Sat Feb 10 11:16:59 2018 by

LaTeXML
diagonal embedding

diagonal embedding

Given a

topological space

X

, the

diagonal embedding

, or

diagonal map

of

X

into

X

×

X

(with the

product topology

) is the map

x

⟼

Δ

(

x

,

x

)

.

X

is

homeomorphic

to the image of

Δ

(which is why we use the word “

embedding

”).

We can perform the same construction with objects other than topological spaces: for instance, there’s a diagonal map

Δ

:

G

→

G

×

G

, from a group into its

direct sum

with itself, given by the same . It’s sensible to call this an embedding, too, since

Δ

is a

monomorphism

.

We could also imagine a diagonal map into an n-fold

product

given by

x

⟼

Δ

n

(

x

,

x

,

…

,

x

)

.

Why call it the diagonal map?

Picture

ℝ

. Its diagonal embedding into the Cartesian plane

ℝ

×

ℝ

is the diagonal line

y

=

x

.

What’s it good for?

Sometimes we can use information about the product space

X

×

X

together with the diagonal embedding to get back information about

X

. For instance,

X

is

Hausdorff

if and only if the image of

Δ

is closed in

X

×

X

[proof (

http://planetmath.org/ASpaceMathnormalXIsHausdorffIfAndOnlyIfDeltaXIsClosed

)]. If we know more about the product space than we do about

X

, it might be easier to check if

Im

⁡

Δ

is closed than to verify the Hausdorff condition directly.

When studying

algebraic topology

, the fact that we have a diagonal embedding for any space

X

lets us define a bit of extra structure in

cohomology

, called the

cup product

. This makes cohomology into a ring, so that we can bring additional algebraic muscle to bear on topological questions.

Another application from algebraic topology: there is something called an

H

-space, which is essentially a topological space in which you can multiply two points together. The diagonal embedding, together with the multiplication, lets us say that the cohomology of an

H

-space is a

Hopf algebra

; this structure lets us find out lots of things about

H

-spaces by

analogy

to what we know about

compact Lie groups

.

Title

diagonal embedding

Canonical name

DiagonalEmbedding

Date of creation

2013-03-22 14:20:41

Last modified on

2013-03-22 14:20:41

Owner

waj (4416)

Last modified by

waj (4416)

Numerical id

8

Author

waj (4416)

Entry type

Definition

Classification

msc 54B10

Classification

msc 18A05

Synonym

diagonal map

Related topic

ASpaceMathnormalXIsHausdorffIfAndOnlyIfDeltaXIsClosed

Generated on Sat Feb 10 11:17:01 2018 by

LaTeXML
product topology and subspace topology

product topology and subspace topology

Let

X

α

with

α

∈

A

be a

collection

of

topological spaces

,
and let

Z

α

⊆

X

α

be subsets. Let

X

=

∏

α

X

α

and

Z

=

∏

α

Z

α

.

In other words,

z

∈

Z

means that

z

is a

function

z

:

A

→

∪

α

Z

α

such that

z

⁢

(

α

)

∈

Z

α

for each

α

. Thus,

z

∈

X

and
we have

Z

⊆

X

as sets.

Theorem 1

.

The

product topology

of

Z

coincides with the

subspace topology

induced by

X

.

Proof.

Let us denote by

τ

X

and

τ

Z

the product topologies for

X

and

Z

, respectively.
Also, let

π

X

,

α

:

X

→

X

α

,

π

Z

,

α

:

Z

→

Z

α

be the

canonical projections

defined for

X

and

Z

.
The

subbases

(

http://planetmath.org/Subbasis

) for

X

and

Z

are given by

β

X

=

{

π

X

,

α

-

1

(

U

)

:

α

∈

A

,

U

∈

τ

(

X

α

)

}

,

β

Z

=

{

π

Z

,

α

-

1

(

U

)

:

α

∈

A

,

U

∈

τ

(

Z

α

)

}

,

where

τ

⁢

(

X

α

)

is the

topology

of

X

α

and

τ

⁢

(

Z

α

)

is the
subspace topology of

Z

α

⊆

X

α

.
The claim follows as

β

Z

=

{

B

∩

Z

:

B

∈

β

X

}

.

∎

Title

product topology and subspace topology

Canonical name

ProductTopologyAndSubspaceTopology

Date of creation

2013-03-22 15:35:33

Last modified on

2013-03-22 15:35:33

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

6

Author

matte (1858)

Entry type

Theorem

Classification

msc 54B10

Generated on Sat Feb 10 11:17:06 2018 by

LaTeXML