aarjaneiro/planetmath huggingface dataset:
Hausdorff space

Hausdorff space

A

topological space

(

X

,

τ

)

is said to be

T

2

(or said to

satisfy

the

T

2

axiom) if given
distinct

x

,

y

∈

X

, there exist

disjoint

open sets

U

,

V

∈

τ

(that is,

U

∩

V

=

∅

)
such that

x

∈

U

and

y

∈

V

.

A

T

2

space is also known as a

Hausdorff space

.
A

Hausdorff topology

for a set

X

is a

topology

τ

such that

(

X

,

τ

)

is a Hausdorff space.

Properties

The following properties are

equivalent

:

1.

X

is a Hausdorff space.

2.

The set

Δ

=

{

(

x

,

y

)

∈

X

×

X

:

x

=

y

}

is closed in the

product topology

of

X

×

X

.

3.

For all

x

∈

X

, we have

{

x

}

=

⋂

{

A

:

A

⊆

X

⁢

closed

,

∃

open set

⁢

U

⁢

such that

⁢

x

∈

U

⊆

A

}

.

Important examples of Hausdorff spaces are

metric spaces

, manifolds,
and

topological vector spaces

.

Title

Hausdorff space

Canonical name

HausdorffSpace

Date of creation

2013-03-22 12:18:18

Last modified on

2013-03-22 12:18:18

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

23

Author

yark (2760)

Entry type

Definition

Classification

msc 54D10

Synonym

Hausdorff topological space

Synonym

T2 space

Related topic

SeparationAxioms

Related topic

T1Space

Related topic

T0Space

Related topic

T3Space

Related topic

RegularSpace

Related topic

MetricSpace

Related topic

NormalTopologicalSpace

Related topic

ASpaceMathnormalXIsHausdorffIfAndOnlyIfDeltaXIsClosed

Related topic

SierpinskiSpace

Related topic

HausdorffSpaceNotCompletelyHausdorff

Related topic

Tychonoff

Related topic

PropertyThatCompactSetsInASpaceAreClosedLies

Defines

Hausdorff

Defines

Hausdorff topology

Defines

T2

Defines

T2 topology

Defines

T2 axiom

Generated on Sat Feb 10 11:21:29 2018 by

LaTeXML
metric spaces are Hausdorff

metric spaces are Hausdorff

Suppose we have a space

X

and a metric

d

on

X

. We’d like to show that the

metric topology

that

d

gives

X

is

Hausdorff

.

Say we’ve got distinct

x

,

y

∈

X

. Since

d

is a metric,

d

⁢

(

x

,

y

)

≠

0

. Then the open balls

B

x

=

B

⁢

(

x

,

d

⁢

(

x

,

y

)

2

)

and

B

y

=

B

⁢

(

y

,

d

⁢

(

x

,

y

)

2

)

are

open sets

in the metric topology which contain

x

and

y

respectively. If we could show

B

x

and

B

y

are

disjoint

, we’d have shown that

X

is Hausdorff.

We’d like to show that an arbitrary point

z

can’t be in both

B

x

and

B

y

. Suppose there is a

z

in both, and we’ll derive a

contradiction

. Since

z

is in these open balls,

d

⁢

(

z

,

x

)

<

d

⁢

(

x

,

y

)

2

and

d

⁢

(

z

,

y

)

<

d

⁢

(

x

,

y

)

2

. But then

d

⁢

(

z

,

x

)

+

d

⁢

(

z

,

y

)

<

d

⁢

(

x

,

y

)

, contradicting the

triangle inequality

.

So

B

x

and

B

y

are disjoint, and

X

is Hausdorff.

□

Title

metric spaces are Hausdorff

Canonical name

MetricSpacesAreHausdorff

Date of creation

2013-03-22 14:21:29

Last modified on

2013-03-22 14:21:29

Owner

waj (4416)

Last modified by

waj (4416)

Numerical id

4

Author

waj (4416)

Entry type

Proof

Classification

msc 54D10

Classification

msc 54E35

Related topic

MetricSpace

Related topic

SeparationAxioms

Generated on Sat Feb 10 11:21:35 2018 by

LaTeXML
proof that a compact set in a Hausdorff space is closed

proof that a compact set in a Hausdorff space is closed

Let

X

be a

Hausdorff space

, and

C

⊆

X

a

compact subset

.
We are to show that

C

is closed.
We will do so, by showing that the

complement

U

=

X

∖

C

is open.
To prove that

U

is open, it suffices to demonstrate that,
for each

x

∈

U

,
there exists an

open set

V

with

x

∈

V

and

V

⊆

U

.

Fix

x

∈

U

.
For each

y

∈

C

, using the

Hausdorff

assumption

,
choose

disjoint

open sets

A

y

and

B

y

with

x

∈

A

y

and

y

∈

B

y

.

Since every

y

∈

C

is an

element of

B

y

,
the

collection

{

B

y

∣

y

∈

C

}

is an open covering of

C

.
Since

C

is

compact

, this

open cover

admits a finite subcover.
So choose

y

1

,

…

,

y

n

∈

C

such that

C

⊆

B

y

1

∪

⋯

∪

B

y

n

.

Notice that

A

y

1

∩

⋯

∩

A

y

n

,
being a finite

intersection

of open sets, is open, and contains

x

.
Call this

neighborhood

of

x

by the name

V

.
All we need to do is show that

V

⊆

U

.

For any point

z

∈

C

, we have

z

∈

B

y

1

∪

⋯

∪

B

y

n

,
and therefore

z

∈

B

y

k

for some

k

.
Since

A

y

k

and

B

y

k

are disjoint,

z

∉

A

y

k

,
and therefore

z

∉

A

y

1

∩

⋯

∩

A

y

n

=

V

.
Thus

C

is disjoint from

V

, and

V

is

contained

in

U

.

Title

proof that a compact set in a Hausdorff space is closed

Canonical name

ProofThatACompactSetInAHausdorffSpaceIsClosed

Date of creation

2013-03-22 13:34:54

Last modified on

2013-03-22 13:34:54

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

7

Author

yark (2760)

Entry type

Proof

Classification

msc 54D10

Classification

msc 54D30

Generated on Sat Feb 10 11:21:37 2018 by

LaTeXML
regular space

regular space

Definition 1

.

A

topological space

is a

regular space

if
it is both a

T

0

space (

http://planetmath.org/T0Space

) and a

T

3

space (

http://planetmath.org/T3Space

).

Example.

Consider the set

ℝ

with the

topology

σ

generated by the basis

β

=

{

U

=

V

-

C

:

V

⁢

is open with the standard topology and

⁢

C

⁢

is (infinite) numerable

}

.

Since

ℚ

is numerable and

ℝ

open, the set of irrational numbers

ℝ

-

ℚ

is open and therefore

ℚ

is closed.
It can be shown that

ℝ

-

ℚ

is an

open set

with this topology and

ℚ

is closed.

Take any irrational number

x

. Any open set

V

containing all

ℚ

must contain also

x

, so the regular space

property

cannot be satisfied. Therefore,

(

ℝ

,

σ

)

is not a regular space.

Note

In topology, the terminology for

separation axioms

is not
standard. Therefore there are also other meanings of

regular

.
In some

references

(e.g.

[

2

]

)
the meanings of regular and

T

3

is exchanged. That is,

T

3

is a

stronger

property than regular.

References

1

L.A. Steen, J.A.Seebach, Jr.,

Counterexamples in topology

,
Holt, Rinehart and Winston, Inc., 1970.

2

J.L. Kelley,

General Topology

, D. van Nostrand Company, Inc., 1955.

Title

regular space

Canonical

name

RegularSpace

Date of creation

2013-03-22 12:18:21

Last modified on

2013-03-22 12:18:21

Owner

drini (3)

Last modified by

drini (3)

Numerical id

11

Author

drini (3)

Entry type

Definition

Classification

msc 54D10

Synonym

regular

Related topic

SeparationAxioms

Related topic

T0Space

Related topic

T2Space

Related topic

T3Space

Related topic

HausdorffSpaceNotCompletelyHausdorff

Related topic

T1Space

Generated on Sat Feb 10 11:21:39 2018 by

LaTeXML
T0 space

T0 space

A

topological space

(

X

,

τ

)

is said to be

T

0

(or to satisfy the

T

0

axiom )
if for all distinct

x

,

y

∈

X

there exists an

open set

U

∈

τ

such that
either

x

∈

U

and

y

∉

U

or

x

∉

U

and

y

∈

U

.

All

T

1

spaces (

http://planetmath.org/T1Space

) are

T

0

.
An example of

T

0

space that is not

T

1

is the

2

-point

Sierpinski space

.

Title

T0 space

Canonical name

T0Space

Date of creation

2013-03-22 12:18:12

Last modified on

2013-03-22 12:18:12

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

13

Author

yark (2760)

Entry type

Definition

Classification

msc 54D10

Synonym

Kolmogorov space

Synonym

Kolmogoroff space

Related topic

Ball

Related topic

T1Space

Related topic

T2Space

Related topic

RegularSpace

Related topic

T3Space

Defines

T0

Generated on Sat Feb 10 11:21:43 2018 by

LaTeXML
T1 space

T1 space

A

topological space

(

X

,

τ

)

is said to be

T

1

(or said to hold the

T

1

axiom) if for all distinct points

x

,

y

∈

X

(

x

≠

y

), there exists an

open set

U

∈

τ

such that

x

∈

U

and

y

∉

U

.

A space being

T

1

is

equivalent

to the following statements:

•

For every

x

∈

X

, the set

{

x

}

is closed.

•

Every subset of

X

is equal to the

intersection

of all the open sets that contain it.

•

Distinct points are

separated

.

Title

T1 space

Canonical

name

T1Space

Date of creation

2013-03-22 12:18:14

Last modified on

2013-03-22 12:18:14

Owner

drini (3)

Last modified by

drini (3)

Numerical id

10

Author

drini (3)

Entry type

Definition

Classification

msc 54D10

Synonym

T1

Related topic

T0Space

Related topic

T2Space

Related topic

T3Space

Related topic

RegularSpace

Related topic

ASpaceIsT1IfAndOnlyIfEverySubsetAIsTheIntersectionOfAllOpenSetsContainingA

Related topic

SierpinskiSpace

Related topic

PropertyThatCompactSetsInASpaceAreClosedLiesStrictlyBetweenT1AndT2

Generated on Sat Feb 10 11:21:44 2018 by

LaTeXML
T3 space

T3 space

A

regular space

is a

topological space

in which points and

closed sets

can be

separated

by

open sets

;
in other words, given a closed set

A

and a point

x

∉

A

,
there are disjoint open sets

U

and

V

such that

x

∈

U

and

A

⊆

V

.

A

T

3

space

is a

regular

T

0

-space (

http://planetmath.org/T0Space

).
A

T

3

space is necessarily also

T

2

, that is,

Hausdorff

.

Note that some authors make the opposite distinction between

T

3

spaces and regular spaces,
that is, they define

T

3

spaces to be topological spaces
in which points and closed sets can be separated by open sets,
and then define regular spaces to be topological spaces
that are both

T

3

and

T

0

.
(With these definitions,

T

3

does not imply

T

2

.)

Title

T3 space

Canonical name

T3Space

Date of creation

2013-03-22 12:18:24

Last modified on

2013-03-22 12:18:24

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

14

Author

yark (2760)

Entry type

Definition

Classification

msc 54D10

Related topic

Tychonoff

Related topic

T2Space

Related topic

T1Space

Related topic

T0Space

Defines

T3

Defines

regular

Defines

regular space

Generated on Sat Feb 10 11:21:46 2018 by

LaTeXML
fully T4

fully T4

A

topological space

X

is said to be

fully

T

4

if every

open cover

of

X

has

star refinement

.

A topological space is said to be

fully normal

if it is a

T

1

space and
is fully

T

4

.

For example, every

pseudometric space

is fully

T

4

.

We have the following

implications

:

Lindelöf

T

3

⇒

paracompact

and

T

3

⇒

fully

T

4

⇒

T

4

⇒

uniformizable

⇒

T

3

,

and

fully normal

⇔

paracompact

regular

.

Title

fully T4

Canonical name

FullyT4

Date of creation

2013-03-22 17:09:43

Last modified on

2013-03-22 17:09:43

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

7

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 54D15

Defines

fully normal

Generated on Sat Feb 10 11:21:54 2018 by

LaTeXML
How are normal and T4 spaces defined in books?

How are normal and T4 spaces defined in books?

A recent discussion on PlanetMath has led me to consider how various
sources define normal and T4 spaces. I limited myself to books, mostly textbooks.
No articles were consulted.
As will be seen from the table below, there is no agreement on
the question of how to define it. I am not giving precise

references

at this time, and may choose to never do that. I think the abbreviated
form may be

sufficient

for those that seek to check what I have done.
If you want to add something to the table, file a correction.
S refers to the condition that

closed sets

can be

separated

by

open sets

.
The condition S is due to Tietze, according to Alexandroff and Hopf.
Of course, T1 + S is the same as T2 +S.

Source

Normal

T4

Page

Year

Alexandroff and Hopf

T1+S

T1+S

68

1935

Wilder

S

?

49

1949

Kelley

S

T1+S

112

1955

Hocking and Young

T1+S

T1+S

41

1961

Pervin

S

T1+S

88

1964

Gaal

T1+S

S

87

1964

Lipschutz

S

T1+S

141

1965

Husain

T1+S

S

7

1966

Dugundji

T2+S

T2+S

144

1966

Gemignani

T1+S

S

102

1967

Willard

S

T1+S

99

1970

Steen and Seebach

T1+S

S

12

1970

Maunder

S

-

15

1970

Munkres

T1+S

-

195

1975

Morris

S

T2+S

121

1988

Repovš

T1+S

S

6

1998

Stroppel

T2+S

S

6

2006

Title

How are normal and T4 spaces defined in books?

Canonical name

HowAreNormalAndT4SpacesDefinedInBooks

Date of creation

2013-03-22 17:09:12

Last modified on

2013-03-22 17:09:12

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

12

Author

Mathprof (13753)

Entry type

Topic

Classification

msc 54D15

Related topic

UrysohnsLemma

Related topic

T4Space

Generated on Sat Feb 10 11:21:55 2018 by

LaTeXML
normal

normal

A

topological space

X

is said to be

normal

if

X

is

T

1

(i.e.

singletons

are closed), and for all disjoint

closed sets

D

,

F

⊆

X

there exist disjoint

open sets

U

and

V

such that

C

⊆

U

and

D

⊆

V

(i.e,

X

is also

T

4

).

Some authors do not require the

T

1

axiom as part of this

definition

.

Title

normal

Canonical

name

Normal

Date of creation

2013-03-22 12:12:39

Last modified on

2013-03-22 12:12:39

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

14

Author

Koro (127)

Entry type

Definition

Classification

msc 54D15

Synonym

normality

Synonym

normal

Related topic

SeparationAxioms

Related topic

Tychonoff

Related topic

Hausdorff

Related topic

CompletelyNormal

Related topic

T2Space

Related topic

AConnectedNormalSpaceWithMoreThanOnePointIsUncountable2

Related topic

AConnectedNormalSpaceWithMoreThanOnePointIsUncountable

Related topic

ApplicationsOfUrysohnsLemmaToLocallyCompactHausdorffSpaces

Generated on Sat Feb 10 11:21:57 2018 by

LaTeXML
T⁢4 space

T

⁢

4

space

Definition 1

.

[

1

]

Suppose

X

is a

topological space

. Further, suppose that
for any two

disjoint

closed sets

A

,

B

⊆

X

, there are two disjoint

open sets

U

and

V

such that

A

⊆

U

and

B

⊆

V

. Then we say
that

X

is a

T

4

space

.

Notes

It should be pointed out that there is no standard convention
for

separation axioms

in

topology

. The above definition follows

[

1

]

. However, in some

references

(e.g.

[

2

]

)
the meaning of

T

4

and normal are exchanged.

References

1

L.A. Steen, J.A.Seebach, Jr.,

Counterexamples in topology

,
Holt, Rinehart and Winston, Inc., 1970.

2

J.L. Kelley,

General Topology

, D. van Nostrand Company, Inc., 1955.

Title

T

⁢

4

space

Canonical name

T4Space

Date of creation

2013-03-22 14:42:15

Last modified on

2013-03-22 14:42:15

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

5

Author

matte (1858)

Entry type

Definition

Classification

msc 54D15

Related topic

SeparationAxioms

Related topic

HowIsNormalityAndT4DefinedInBooks

Generated on Sat Feb 10 11:22:01 2018 by

LaTeXML
Tietze extension theorem

Tietze extension theorem

Let

X

be a

topological space

. Then the following are

equivalent

:

1.

X

is normal.

2.

If

A

is a

closed subset

in

X

, and

f

:

A

→

[

-

1

,

1

]

is a

continuous function

, then

f

has a

continuous

to all of

X

.
(In other words, there is a continuous function

f

∗

:

X

→

[

-

1

,

1

]

such that

f

and

f

∗

coincide on

A

.)

Remark:

If

X

and

A

are as above, and

f

:

A

→

(

-

1

,

1

)

is a continuous function, then

f

has a continuous to all of

X

.

The present result can be found in

[

1

]

.

References

1

A. Mukherjea, K. Pothoven,

Real and

Functional analysis

,
Plenum press, 1978.

Title

Tietze extension theorem

Canonical name

TietzeExtensionTheorem

Date of creation

2013-03-22 13:35:30

Last modified on

2013-03-22 13:35:30

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

5

Author

matte (1858)

Entry type

Theorem

Classification

msc 54D15

Related topic

ApplicationsOfUrysohnsLemmaToLocallyCompactHausdorffSpaces

Generated on Sat Feb 10 11:22:03 2018 by

LaTeXML
Tychonoff space

Tychonoff space

A

topological space

X

is said to be

completely regular

if whenever

C

⊆

X

is closed and

x

∈

X

∖

C

then there is a

continuous function

f

:

X

→

[

0

,

1

]

with

f

⁢

(

x

)

=

0

and

f

⁢

(

C

)

⊆

{

1

}

.

A completely regular space that is also

T

0

(

http://planetmath.org/T0Space

) (and therefore

Hausdorff

(

http://planetmath.org/T2Space

))
is called a

Tychonoff space

, or a

T

3

⁤

1

2

space

.

Some authors interchange the meanings of ‘completely regular’ and ‘

T

3

⁤

1

2

’ compared to the above.

It can be proved that a topological space is Tychonoff
if and only if it has a

Hausdorff compactification

.

Title

Tychonoff space

Canonical

name

TychonoffSpace

Date of creation

2013-03-22 12:12:42

Last modified on

2013-03-22 12:12:42

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

11

Author

yark (2760)

Entry type

Definition

Classification

msc 54D15

Synonym

Tikhonov space

Synonym

Tychonoff topological space

Synonym

Tikhonov topological space

Synonym

Tychonov space

Synonym

Tychonov topological space

Related topic

NormalTopologicalSpace

Related topic

T3Space

Defines

Tychonoff

Defines

completely regular

Defines

completely regular space

Defines

Tikhonov

Defines

Tychonov

Generated on Sat Feb 10 11:22:04 2018 by

LaTeXML
Urysohn’s lemma

Urysohn’s lemma

A

normal space

is a

topological space

X

such that whenever

A

and

B

are

disjoint

closed subsets

of

X

,
then there are disjoint

open subsets

U

and

V

of

X

such that

A

⊆

U

and

B

⊆

V

.

(Note that some authors include

T

1

in the

definition

,
which is

equivalent

to requiring the space to be

Hausdorff

.)

Urysohn’s Lemma

states that

X

is normal
if and only if
whenever

A

and

B

are disjoint closed subsets of

X

,
then there is a

continuous function

f

:

X

→

[

0

,

1

]

such that

f

⁢

(

A

)

⊆

{

0

}

and

f

⁢

(

B

)

⊆

{

1

}

.
(Any such

function

is called an

Urysohn function

.)

A

corollary

of Urysohn’s Lemma
is that normal

T

1

(

http://planetmath.org/T1Space

) spaces are

completely regular

.

Title

Urysohn’s lemma

Canonical

name

UrysohnsLemma

Date of creation

2013-03-22 12:12:34

Last modified on

2013-03-22 12:12:34

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

12

Author

yark (2760)

Entry type

Theorem

Classification

msc 54D15

Related topic

HowIsNormalityAndT4DefinedInBooks

Related topic

ApplicationsOfUrysohnsLemmaToLocallyCompactHausdorffSpaces

Defines

Urysohn function

Defines

normal space

Defines

normal topological space

Defines

normal

Defines

normality

Generated on Sat Feb 10 11:22:06 2018 by

LaTeXML
collectionwise normal

collectionwise normal

A

Hausdorff topological space

X

is called

collectionwise normal

if any

discrete

collection

of sets

{

U

i

}

in

X

can be covered by a pairwise-disjoint collection of

open sets

{

V

j

}

such that each

V

j

covers just one

U

i

. This is

equivalent

to requiring the same property for any discrete collection of

closed sets

.

A Hausdorff topological space

X

is called

countably collectionwise normal

if any

countable

discrete collection of sets

{

U

i

}

in

X

can be covered by a pairwise-disjoint collection of open sets

{

V

j

}

such that each

V

j

covers just one

U

i

. This is equivalent to requiring the same property for any countable discrete collection of closed sets.

Any

metrizable space

is collectionwise normal.

References

1

Steen, Lynn Arthur and Seebach, J. Arthur,

Counterexamples in Topology

, Dover Books, 1995.

Title

collectionwise normal

Canonical name

CollectionwiseNormal

Date of creation

2013-03-22 14:49:54

Last modified on

2013-03-22 14:49:54

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

5

Author

mathcam (2727)

Entry type

Definition

Classification

msc 54D20

Defines

countably collectionwise normal

Generated on Sat Feb 10 11:22:07 2018 by

LaTeXML
countably compact

countably compact

A

topological space

X

is said to be

countably compact

if every

countable

open cover

has a finite subcover.

Countable compactness is

equivalent

to

limit point compactness

if

A

is

T

1

spaces, and is equivalent to

compactness

(

http://planetmath.org/

Compact

) if

X

is a

metric space

.

Title

countably compact

Canonical name

CountablyCompact

Date of creation

2013-03-22 12:06:43

Last modified on

2013-03-22 12:06:43

Owner

Evandar (27)

Last modified by

Evandar (27)

Numerical id

8

Author

Evandar (27)

Entry type

Definition

Classification

msc 54D20

Synonym

countable compactness

Related topic

Compact

Related topic

Lindelof

Related topic

LimitPointCompact

Generated on Sat Feb 10 11:22:09 2018 by

LaTeXML
development

development

Let

X

be a

topological space

. A

development

for

X

is a

countable

collection

F

1

,

F

2

,

…

of open coverings of

X

such that for any

closed subset

C

of

X

and any point

p

in the

complement

of

C

, there exists a cover

F

j

such that no element of

F

j

which contains

p

intersects

C

. A space with a development is called

developable

.

A development

F

1

,

F

2

,

…

such that

F

i

⊂

F

i

+

1

for all

i

is called a

nested development

. A

theorem

from Vickery states that every developable space in fact has a nested development.

References

1

Steen, Lynn Arthur and Seebach, J. Arthur,

Counterexamples in Topology

, Dover Books, 1995.

Title

development

Canonical

name

Development

Date of creation

2013-03-22 14:49:49

Last modified on

2013-03-22 14:49:49

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Definition

Classification

msc 54D20

Defines

developable

Defines

nested development

Defines

Vickery’s theorem

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

LaTeXML
existence of partitions of unity

existence of partitions of unity

Theorem -

Let

X

be a

paracompact

Hausdorff space

and

{

U

i

}

i

∈

J

an open covering of

X

. Then there exists a

partition of unity

on

X

subordinate to

{

U

i

}

i

∈

J

.

Title

existence of partitions of unity

Canonical name

ExistenceOfPartitionsOfUnity

Date of creation

2013-03-22 17:24:29

Last modified on

2013-03-22 17:24:29

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

5

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 54D20

Generated on Sat Feb 10 11:22:12 2018 by

LaTeXML
Lindelöf space

Lindelöf space

Definition

A

topological space

is said to be

Lindelöf

if every

open cover

has a

countable

subcover.

Notes

A second-countable space is Lindelöf.
A

compact space

is Lindelöf.

A

regular

(

http://planetmath.org/T3Space

) Lindelöf space is

http://planetmath.org/node/1530

normal.

F

σ

sets (

http://planetmath.org/F_sigmaSet

) in Lindelöf spaces are Lindelöf.

Continuous

images of Lindelöf spaces are Lindelöf.

A Lindelöf space is

compact

if and only if it is

countably compact

.

Title

Lindelöf space

Canonical

name

LindelofSpace

Date of creation

2013-03-22 12:06:34

Last modified on

2013-03-22 12:06:34

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

11

Author

yark (2760)

Entry type

Definition

Classification

msc 54D20

Related topic

SecondCountable

Related topic

Separable

Related topic

Compact

Related topic

LindelofTheorem

Related topic

CompactMetricSpacesAreSecondCountable

Related topic

ErnstLindelof

Defines

Lindelöf

Defines

Lindelöf

property

Generated on Sat Feb 10 11:22:13 2018 by

LaTeXML