aarjaneiro/planetmath huggingface dataset:
proof of Tychonoff’s theorem in finite case

proof of Tychonoff’s theorem in finite case

(The finite case of

Tychonoff

’s

Theorem

is of course a subset of the

infinite

case,
but the proof is substantially easier, so that is why it is presented here.)

To prove that

X

1

×

⋯

×

X

n

is

compact

if the

X

i

are compact, it suffices (by

induction

) to prove that

X

×

Y

is compact
when

X

and

Y

are. It also suffices to prove that
a finite

subcover

can be extracted from every open cover of

X

×

Y

by only the

basis sets

of the form

U

×

V

, where

U

is open in

X

and

V

is open in

Y

.

Proof.

The proof is by the straightforward strategy of composing a finite subcover
from a lower-dimensional subcover. Let the open cover

𝒞

of

X

×

Y

by basis sets be given.

The set

X

×

{

y

}

is compact, because it is the image of a

continuous

embedding

of the compact set

X

.
Hence

X

×

{

y

}

has a finite subcover in

𝒞

: label the subcover
by

𝒮

y

=

{

U

1

y

×

V

1

y

,

…

,

U

k

y

y

×

V

k

y

y

}

.
Do this for each

y

∈

Y

.

To get the desired subcover of

X

×

Y

, we need to pick a finite number
of

y

∈

Y

. Consider

V

y

=

⋂

i

=

1

k

y

V

i

y

.
This is a finite

intersection

of

open sets

, so

V

y

is open in

Y

.
The

collection

{

V

y

:

y

∈

Y

}

is an open

covering

of

Y

, so pick
a finite subcover

V

y

1

,

…

,

V

y

l

.
Then

⋃

j

=

1

l

𝒮

y

j

is a finite subcover
of

X

×

Y

.
∎

Title

proof of Tychonoff’s theorem in finite case

Canonical name

ProofOfTychonoffsTheoremInFiniteCase

Date of creation

2013-03-22 15:26:27

Last modified on

2013-03-22 15:26:27

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

4

Author

stevecheng (10074)

Entry type

Proof

Classification

msc 54D30

Generated on Sat Feb 10 11:22:48 2018 by

LaTeXML
properties of compact spaces

properties of compact spaces

We list the following commonly quoted properties of

compact

topological spaces

.

•

A

closed subset

of a

compact space

is compact

•

A compact

subspace

of a

Hausdorff space

is closed

•

The

continuous

image of a compact space is compact

•

Compactness

is

equivalent

to

sequential compactness

in the

metric topology

Title

properties of compact spaces

Canonical name

PropertiesOfCompactSpaces

Date of creation

2013-03-22 13:25:07

Last modified on

2013-03-22 13:25:07

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

7

Author

rspuzio (6075)

Entry type

Result

Classification

msc 54D30

Generated on Sat Feb 10 11:22:50 2018 by

LaTeXML
pseudocompact space

pseudocompact space

A

topological space

X

is said to be

pseudocompact

if every

continuous function

f

:

X

→

ℝ

has

bounded

image.

All countably compact spaces (which includes all

compact spaces

and all sequentially compact spaces) are pseudocompact.
A

metric space

is pseudocompact if and only if it is

compact

.
A

Hausdorff

normal space

is pseudocompact if and only if it is

countably compact

.

Title

pseudocompact space

Canonical name

PseudocompactSpace

Date of creation

2013-03-22 14:20:36

Last modified on

2013-03-22 14:20:36

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

7

Author

yark (2760)

Entry type

Definition

Classification

msc 54D30

Synonym

pseudo compact space

Synonym

pseudo-compact space

Related topic

LimitPointCompact

Defines

pseudocompact

Defines

pseudocompactness

Defines

pseudo-compact

Defines

pseudo-compactness

Defines

pseudo compact

Defines

pseudo compactness

Generated on Sat Feb 10 11:22:51 2018 by

LaTeXML
representation theorem for compact metric spaces

representation theorem for compact metric spaces

Theorem

.

Every

compact

metric space

is a

continuous

image of the Cantor set.

References

1

S. Willard,

General Topology

,
Addison-Wesley, Publishing Company, 1970.

Title

representation theorem for compact metric spaces

Canonical name

RepresentationTheoremForCompactMetricSpaces

Date of creation

2013-03-22 14:40:02

Last modified on

2013-03-22 14:40:02

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

8

Author

matte (1858)

Entry type

Theorem

Classification

msc 54D30

Classification

msc 54E35

Related topic

Dimension3

Related topic

CantorSet

Related topic

HahnMazurkiewiczTheorem

Generated on Sat Feb 10 11:22:53 2018 by

LaTeXML
Stone-Čech compactification

Stone-Čech compactification

Stone-Čech

compactification

is a technique for

embedding

a

Tychonoff topological space

in a

compact

Hausdorff space

.

Let

X

be a

Tychonoff space

and let

C

be the space of all

continuous functions

from

X

to the

closed interval

[

0

,

1

]

. To each

element

x

∈

X

, we may associate the evaluation

functional

e

x

:

C

→

[

0

,

1

]

defined by

e

x

⁢

(

f

)

=

f

⁢

(

x

)

. In this way,

X

may be identified with a set of functionals.

The space

[

0

,

1

]

C

of

all

functionals from

C

to

[

0

,

1

]

may be endowed with the

Tychonoff product topology

.

Tychonoff

’s

theorem

asserts that, in this

topology

,

[

0

,

1

]

C

is a compact Hausdorff space. The

closure

in this topology of the subset of

[

0

,

1

]

C

which was identified with

X

via evaluation functionals is

β

⁢

X

, the Stone-Čech compactification of

X

.
Being a

closed subset

of a compact Hausdorff space,

β

⁢

X

is itself a compact Hausdorff space.

This construction has the wonderful

property

that, for any compact Hausdorff space

Y

, every continuous function

f

:

X

→

Y

may be extended to a

unique

continuous function

β

⁢

f

:

β

⁢

X

→

Y

.

Title

Stone-Čech compactification

Canonical

name

StonevCechCompactification

Date of creation

2013-03-22 14:37:38

Last modified on

2013-03-22 14:37:38

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

10

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 54D30

Generated on Sat Feb 10 11:22:54 2018 by

LaTeXML
the continuous image of a compact space is compact

the continuous image of a compact space is compact

Consider

f

:

X

→

Y

a

continuous

and surjective function and

X

a

compact set

. We will prove that

Y

is also a compact set.

Let

{

V

a

}

be an open covering of

Y

. By the

continuity

of

f

the pre-image by

f

of any

open subset

(

http://planetmath.org/OpenSubset

) of

Y

will also be an open subset of

X

.
So we have an open covering

{

U

a

}

of

X

where

U

a

=

f

-

1

⁢

(

V

a

)

.

To see this remember, since the continuity of

f

implies that each

U

a

is open, all we need to prove is that

⋃

a

U

a

⊃

X

.
Consider

x

∈

X

, we know that since

{

V

a

}

is a covering of

Y

that there exists

i

such that

f

⁢

(

x

)

∈

V

i

but then by construction

x

∈

U

i

and

{

U

a

}

is indeed an open covering of

X

.

Since

X

is

compact

we can consider a

finite set

of indices

{

a

i

}

such that

{

U

a

i

}

is a finite open covering of

X

, but then

{

V

a

i

}

will be a finite open covering of

Y

and it will thus be a compact set.

To see that

{

V

a

i

}

is a covering of

Y

consider

y

∈

Y

. By the surjectivity of

f

there must exist (at least) one

x

∈

X

such that

f

⁢

(

x

)

=

y

and since

{

U

a

i

}

is a finite covering of

X

, there exists

k

such that

x

∈

U

a

k

.
But then since

f

⁢

(

U

a

k

)

=

V

a

k

, we must have that

y

∈

V

a

k

and

{

V

a

i

}

is indeed a finite open covering of

Y

.

Title

the continuous image of a

compact space

is compact

Canonical name

TheContinuousImageOfACompactSpaceIsCompact

Date of creation

2013-03-22 15:52:48

Last modified on

2013-03-22 15:52:48

Owner

cvalente (11260)

Last modified by

cvalente (11260)

Numerical id

7

Author

cvalente (11260)

Entry type

Proof

Classification

msc 54D30

Related topic

CompactnessIsPreservedUnderAContinuousMap

Generated on Sat Feb 10 11:22:56 2018 by

LaTeXML
The property that compact sets in a space are closed lies strictly between T1 and T2

The property that compact sets in a space are closed lies strictly between T1 and T2

If a

topological space

is

Hausdorff

(

T

2

), then every

compact subset

of that space is closed. If every compact subset of a space is closed, then (since singletons are always

compact

) then the space is

accessible

(

T

1

). There are spaces that are

T

1

and have compact sets that are not closed, and there are spaces in which compact sets are always closed but that are not

T

2

.

Let

X

be an

infinite set

with the finite

complement

topology

. Singletons are finite, and therefore closed, so

X

is

T

1

. Let

S

⊂

X

, and let

𝔽

be an

open cover

of

S

. Let

F

∈

𝔽

. Then

X

∖

F

is finite. Choosing a member of

𝔽

for each remaining element of

S

shows that

𝔽

has a finite subcover. Thus, every subset of

X

is compact. An infinite subset of

X

will then be compact, but not closed.

Let

Y

be an

uncountable set

with the countable complement topology. No two

open sets

are

disjoint

, so

Y

is not Hausdorff. Let

C

be a compact subset of

Y

. I shall show that

C

is finite. Suppose

C

is

infinite

, and let

S

be an infinite

sequence

in

C

without any repetitions. For any

natural number

n

, let

U

n

be all the elements of

C

except for all the

S

k

, where

k

>

n

. Then

U

n

is open for each

n

, and

{

U

n

∣

n

∈

ℕ

}

covers

C

, but has no finite subset that covers

C

, contradicting the fact that

C

is compact. This

contradiction

arose by assuming a compact subset of

Y

was infinite, all compact subsets of

Y

are finite.

Y

is

T

1

(

singleton sets

are

countable

), so all compact subsets of

Y

are closed.

These examples were suggested by the person known as Polytope on EFNet’s math channel.

Title

The property that compact sets in a space are closed lies strictly between T1 and T2

Canonical name

ThePropertyThatCompactSetsInASpaceAreClosedLiesStrictlyBetweenT1AndT2

Date of creation

2013-03-22 17:38:59

Last modified on

2013-03-22 17:38:59

Owner

dfeuer (18434)

Last modified by

dfeuer (18434)

Numerical id

5

Author

dfeuer (18434)

Entry type

Result

Classification

msc 54D30

Classification

msc 54D10

Related topic

T1Space

Related topic

T2Space

Related topic

Compact

Generated on Sat Feb 10 11:22:58 2018 by

LaTeXML
tube lemma

tube lemma

Tube lemma

-

Let

X

and

Y

be

topological spaces

such that

Y

is

compact

. If

N

is an

open set

of

X

×

Y

containing a ”slice”

x

0

×

Y

, then

N

contains some ”tube”

W

×

Y

, where

W

is a

neighborhood

of

x

0

in

X

.

Proof :

N

is a union of basis elements

U

×

V

, with

U

and

V

open sets in

X

and

Y

respect. Since

x

0

×

Y

is compact (it is

homeomorphic

to

Y

), only a finite number

U

1

×

V

1

,

…

,

U

n

×

V

n

of such basis elements cover

x

0

×

Y

.

We may assume that each of the basis elements

U

i

×

V

i

actually

intersects

x

0

×

Y

, since otherwise we could discard it from the finite

collection

and still have a covering of

x

0

×

Y

.

Define

W

:=

U

1

∩

…

∩

U

n

. The set

W

is open and contains

x

0

because each

U

i

×

V

i

intersects

x

0

×

Y

by the previous remark.

We now claim that

W

×

Y

⊆

N

. Let

(

x

,

y

)

be a point in

W

×

Y

. The point

(

x

0

,

y

)

is in some

U

i

×

V

i

and so

y

∈

V

i

. We also know that

x

∈

W

=

U

1

∩

…

∩

U

n

⊆

U

i

.

Therefore

(

x

,

y

)

∈

U

i

×

V

i

⊆

N

as desired.

□

References

1

J. Munkres,

Topology

(2nd edition), Prentice Hall, 1999.

Title

tube lemma

Canonical name

TubeLemma

Date of creation

2013-03-22 17:25:39

Last modified on

2013-03-22 17:25:39

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

7

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 54D30

Generated on Sat Feb 10 11:23:00 2018 by

LaTeXML
Tychonoff’s theorem

Tychonoff’s theorem

Let

(

X

i

)

i

∈

I

be a family of nonempty

topological spaces

. The product space (see

product topology

)

∏

i

∈

I

X

i

is

compact

if and only if each of the spaces

X

i

is compact.

Not surprisingly, if

I

is

infinite

, the proof requires the

Axiom of Choice

.

Conversely

, one can show that

Tychonoff

’s

theorem

implies that any

product

of

nonempty sets

is nonempty, which is one form of the Axiom of Choice.

Title

Tychonoff’s theorem

Canonical name

TychonoffsTheorem

Date of creation

2013-03-22 12:05:14

Last modified on

2013-03-22 12:05:14

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

12

Author

matte (1858)

Entry type

Theorem

Classification

msc 54D30

Synonym

Tichonov’s theorem

Related topic

Compact

Generated on Sat Feb 10 11:23:01 2018 by

LaTeXML
weakly countably compact

weakly countably compact

A

topological space

X

is said to be

weakly countably compact

(or

limit point compact

)
if every

infinite subset

of

X

has a

limit point

.

Every countably compact space is weakly countably compact.
The

converse

is true in

T

1

spaces (

http://planetmath.org/T1Space

).

A

metric space

is weakly countably compact if and only if it is

compact

.

An easy example of a space

X

that is not weakly countably compact
is any

infinite set

with the

discrete topology

.
A more interesting example is the countable complement topology
on an

uncountable set

.

Title

weakly countably compact

Canonical name

WeaklyCountablyCompact

Date of creation

2013-03-22 12:06:46

Last modified on

2013-03-22 12:06:46

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

9

Author

yark (2760)

Entry type

Definition

Classification

msc 54D30

Synonym

limit point compact

Synonym

limit-point compact

Related topic

Compact

Related topic

CountablyCompact

Related topic

SequentiallyCompact

Related topic

PseudocompactSpace

Defines

limit point compactness

Defines

weak countable compactness

Generated on Sat Feb 10 11:23:12 2018 by

LaTeXML
Y is compact if and only if every open cover of Y has a finite subcover

Y

is compact if and only if every open cover of

Y

has a finite subcover

Theorem

.

Let

X

be a

topological space

and

Y

a subset of

X

. Then the following
statements are

equivalent

.

1.

Y

is

compact

as a subset of

X

.

2.

Every

open cover

of

Y

(with

open sets

in

X

) has a finite
subcover.

Proof.

Suppose

Y

is compact, and

{

U

i

}

i

∈

I

is an arbitrary
open cover of

Y

, where

U

i

are open sets in

X

. Then

{

U

i

∩

Y

}

i

∈

I

is a

collection

of open sets in

Y

with
union

Y

. Since

Y

is compact, there is a finite subset

J

⊂

I

such that

Y

=

∪

i

∈

J

(

U

i

∩

Y

)

. Now

Y

=

(

∪

i

∈

J

U

i

)

∩

Y

⊂

∪

i

∈

J

U

i

, so

{

U

i

}

i

∈

J

is finite open cover of

Y

.

Conversely

, suppose every open cover of

Y

has a finite subcover,
and

{

U

i

}

i

∈

I

is an arbitrary collection of open sets
(in

Y

) with union

Y

. By the definition of the

subspace topology

,
each

U

i

is of the form

U

i

=

V

i

∩

Y

for some open set

V

i

in

X

. Now

U

i

⊂

V

i

, so

{

V

i

}

i

∈

I

is a cover
of

Y

by open sets in

X

. By

assumption

, it has a finite subcover

{

V

i

}

i

∈

J

. It follows that

{

U

i

}

i

∈

J

covers

Y

, and

Y

is compact.

□

The above proof follows the proof given in

[

1

]

.

References

1

B.Ikenaga,

Notes on

Topology

, August 16, 2000, available online

http://www.millersv.edu/ bikenaga/topology/topnote.html

http://www.millersv.edu/ bikenaga/topology/topnote.html.

Title

Y

is compact if and only if every open cover of

Y

has a finite subcover

Canonical name

YIsCompactIfAndOnlyIfEveryOpenCoverOfYHasAFiniteSubcover

Date of creation

2013-03-22 13:34:07

Last modified on

2013-03-22 13:34:07

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 54D30

Generated on Sat Feb 10 11:23:14 2018 by

LaTeXML
Alexandrov one-point compactification

Alexandrov one-point compactification

The

Alexandrov one-point compactification

of a non-compact

topological space

X

is obtained by adjoining a new point

∞

and defining the

topology

on

X

∪

{

∞

}

to consist of the

open sets

of

X

together with the sets of the form

U

∪

{

∞

}

, where

U

is an open subset of

X

with

compact

complement.

With this topology,

X

∪

{

∞

}

is always compact.
Furthermore, it is

Hausdorff

if and only if

X

is Hausdorff and

locally compact

.

Title

Alexandrov one-point compactification

Canonical name

AlexandrovOnepointCompactification

Date of creation

2013-03-22 13:47:54

Last modified on

2013-03-22 13:47:54

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

9

Author

yark (2760)

Entry type

Definition

Classification

msc 54D35

Synonym

one-point compactification

Synonym

Alexandroff one-point compactification

Synonym

Aleksandrov one-point compactification

Synonym

Alexandrov compactification

Synonym

Aleksandrov compactification

Synonym

Alexandroff compactification

Related topic

Compactification

Generated on Sat Feb 10 11:23:16 2018 by

LaTeXML
compactification

compactification

Let

X

be a

topological space

. A (

Hausdorff

)

compactification

of

X

is a pair

(

K

,

h

)

where

K

is a Hausdorff topological space and

h

:

X

→

K

is a

continuous function

such that

•

K

is

compact

•

h

is a

homeomorphism

between

X

and

h

⁢

(

X

)

•

h

⁢

(

X

)

¯

K

=

K

where

A

¯

K

denotes

closure

in

K

for any subset

A

of

K

h

is often considered to be the

inclusion map

, so that

X

⊆

K

with

X

¯

K

=

K

.

Title

compactification

Canonical name

Compactification

Date of creation

2013-03-22 12:15:42

Last modified on

2013-03-22 12:15:42

Owner

Evandar (27)

Last modified by

Evandar (27)

Numerical id

8

Author

Evandar (27)

Entry type

Definition

Classification

msc 54D35

Synonym

Hausdorff compactification

Related topic

Compact

Related topic

AlexandrovOnePointCompactification

Generated on Sat Feb 10 11:23:17 2018 by

LaTeXML
completion

completion

Let

(

X

,

d

)

be a

metric space

. Let

X

¯

be the set of all

Cauchy
sequences

{

x

n

}

n

∈

ℕ

in

X

. Define an

equivalence relation

∼

on

X

¯

by setting

{

x

n

}

∼

{

y

n

}

if the

interleave sequence

of the

sequences

{

x

n

}

and

{

y

n

}

is also a
Cauchy sequence. The

completion

of

X

is defined to be the set

X

^

of

equivalence classes

of

X

¯

modulo

∼

.

The metric

d

on

X

extends to a metric on

X

^

in the
following manner:

d

⁢

(

{

x

n

}

,

{

y

n

}

)

:=

lim

n

→

∞

⁡

d

⁢

(

x

n

,

y

n

)

,

where

{

x

n

}

and

{

y

n

}

are representative Cauchy sequences of

elements

in

X

^

. The definition of

∼

is tailored so that
the limit in the above definition is well defined, and the fact that these
sequences are Cauchy, together with the fact that

ℝ

is

complete

,
ensures that the limit exists. The space

X

^

with this metric is of course a

complete metric space

.

The original metric space

X

is

isometric

to the subset of

X

^

consisting of equivalence classes of

constant

sequences.

Note the similarity between the construction of

X

^

and the
construction of

ℝ

from

ℚ

. The process used here is the same as
that used to

construct

the

real numbers

ℝ

, except for the minor
detail that one can not use the terminology of metric spaces in the
construction of

ℝ

itself because it is necessary to construct

ℝ

in the first place before one can define metric spaces.

1

Metric spaces with richer structure

If the metric space

X

has an

algebraic structure

, then in many
cases this algebraic structure carries through unchanged to

X

^

simply by applying it one element at a time to sequences in

X

. We
will not attempt to state this principle precisely, but we will
mention the following important

instances

:

1.

If

(

X

,

⋅

)

is a topological group, then

X

^

is also a
topological group with

multiplication

defined by

{

x

n

}

⋅

{

y

n

}

=

{

x

n

⋅

y

n

}

.

2.

If

X

is a

topological ring

, then

addition

and multiplication
extend to

X

^

and make the completion into a topological ring.

3.

If

F

is a field with a

valuation

v

, then the completion of

F

with respect to the metric imposed by

v

is a topological
field, denoted

F

v

and called the completion of

F

at

v

.

2

Universal property of completions

The completion

X

^

of

X

satisfies

the following

universal property

: for every uniformly continuous map

f

:

X

⟶

Y

of

X

into a complete metric space

Y

, there exists a unique lifting of

f

to a

continuous map

f

^

:

X

^

⟶

Y

making the

diagram

Generated on Sat Feb 10 11:23:22 2018 by

LaTeXML
something related to Alexandrov one-point compactification

something related to Alexandrov one-point compactification

The

topology

for

X

⁢

⋃

{

∞

}

is defined as follows: there are two kinds of

open sets

of

X

⁢

⋃

{

∞

}

. If

∞

∉

U

, then

U

is open if and only if

U

is an open set in the topology of

X

. If

∞

∈

U

, then

U

is open if and only if

U

is the complement of a closed

compact subset

K

of

X

.

Title

something related to

Alexandrov one-point compactification

Canonical name

SomethingRelatedToAlexandrovOnepointCompactification

Date of creation

2013-03-22 17:03:54

Last modified on

2013-03-22 17:03:54

Owner

adrianita (17056)

Last modified by

adrianita (17056)

Numerical id

4

Author

adrianita (17056)

Entry type

Definition

Classification

msc 54D35

Generated on Sat Feb 10 11:23:24 2018 by

LaTeXML
every σ-compact set is Lindelöf

every

σ

-compact set is Lindelöf

Theorem 1

.

Every

σ

-compact (

http://planetmath.org/SigmaCompact

) set is Lindelöf (every open cover has a

countable

subcover).

Proof.

Let

X

be a

σ

-compact. Let

𝒜

be an open cover of

X

. Since

X

is

σ

-compact, it is the union of countable
many compact sets,

X

=

⋃

i

=

0

∞

X

i

with

X

i

compact. Consider the cover

𝒜

i

=

{

A

∈

𝒜

:

X

i

∩

A

≠

∅

}

of the set

X

i

. This cover is well defined, it is not empty and covers

X

i

: for each

x

∈

X

i

there is at least one of the open sets

A

∈

𝒜

such that

x

∈

A

.

Since

X

i

is compact, the cover

𝒜

i

has a finite subcover. Then

X

i

⊆

⋃

j

=

0

N

j

A

i

j

and thus

X

⊆

⋃

i

=

0

∞

(

⋃

j

=

0

N

j

A

i

j

)

.

That is, the set

{

A

i

j

}

is a countable subcover of

𝒜

that covers

X

.
∎

Title

every

σ

-compact set is Lindelöf

Canonical name

EverysigmacompactSetIsLindelof

Date of creation

2013-03-22 17:34:07

Last modified on

2013-03-22 17:34:07

Owner

joen235 (18354)

Last modified by

joen235 (18354)

Numerical id

14

Author

joen235 (18354)

Entry type

Theorem

Classification

msc 54D45

Generated on Sat Feb 10 11:23:26 2018 by

LaTeXML