aarjaneiro/planetmath huggingface dataset:
Tychonoff fixed point theorem

Tychonoff fixed point theorem

Let

X

be a

locally convex topological vector space

, and let

K

⊂

X

be a non-empty,

compact

, and convex set.
Then given any

continuous mapping

f

:

K

→

K

there exists

x

∈

K

such that

f

⁢

(

x

)

=

x

.

Notice that a

normed vector space

is a locally convex topological
vector space so this theorem extends the

Schauder fixed point theorem

.

References

1

Rudin,

Functional Analysis

, Chapter 5.

Title

Tychonoff fixed point theorem

Canonical name

TychonoffFixedPointTheorem

Date of creation

2013-03-22 16:04:11

Last modified on

2013-03-22 16:04:11

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

8

Author

paolini (1187)

Entry type

Theorem

Classification

msc 54H25

Classification

msc 46B50

Classification

msc 47H10

Related topic

SchauderFixedPointTheorem

Related topic

BrouwerFixedPointTheorem

Generated on Sat Feb 10 11:31:44 2018 by

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graph topology

graph topology

A graph

(

V

,

E

)

is identified by its vertices

V

=

{

v

1

,

v

2

,

…

}

and its
edges

E

=

{

{

v

i

,

v

j

}

,

{

v

k

,

v

l

}

,

…

}

. A graph also admits a natural
topology, called the

graph topology

, by identifying every edge

{

v

i

,

v

j

}

with the unit interval

I

=

[

0

,

1

]

and gluing them together at
coincident vertices.

This construction can be easily realized in the

framework

of

simplicial
complexes

. We can form a simplicial complex

G

=

{

{

v

}

∣

v

∈

V

}

∪

E

. And the desired topological realization of the graph is just the

geometric realization

|

G

|

of

G

.

Viewing a graph as a

topological space

has several advantages:

•

The notion of

graph isomorphism

becomes that of simplicial (or cell) complex (

http://planetmath.org/CWComplex

)

isomorphism

.

•

The notion of a

connected graph

coincides with topological

connectedness

(

http://planetmath.org/ConnectedSpace

).

•

A connected graph is a tree if and only if its

fundamental group

is trivial.

Remark:

A graph is/can be regarded as a one-dimensional

C

⁢

W

-complex.

Title

graph topology

Canonical name

GraphTopology

Date of creation

2013-03-22 13:37:03

Last modified on

2013-03-22 13:37:03

Owner

mps (409)

Last modified by

mps (409)

Numerical id

10

Author

mps (409)

Entry type

Definition

Classification

msc 54H99

Classification

msc 05C62

Classification

msc 05C10

Synonym

one-dimensional CW complex

Related topic

GraphTheory

Related topic

Graph

Related topic

ConnectedGraph

Related topic

QuotientSpace

Related topic

Realization

Related topic

RSupercategory

Related topic

CWComplexDefinitionRelatedToSpinNetworksAndSpinFoams

Generated on Sat Feb 10 11:31:46 2018 by

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Background of several open problems I stumbled upon

Background of several open problems I stumbled upon

This lists some problem aroused while I wrote my book “Algebraic General Topology. Volume 1”.

The draft of the book is now

\href

http://www.mathematics21.org/algebraic-general-topology.htmlavailable online.

Please first read the book and then collaborate with me in solving the below listed problems.

Theorems

to prove:

•

Intersection

with a staroid

product

through its upper sets.

•

Intersecting two staroidal products

.

•

Multifuncoid has atomic arguments

.

•

Every

regular

paratopological group is

Tychonoff

.

More related materials:

•

Direct products

in a

category

of funcoids.

•

Compact

funcoids.

•

Meta-singular numbers.

See the right sidebar of this page (”Attached articles”) for the list of problems.

TODO: More information should be put to this page.

See also

\href

http://researchtrends.wikia.com/wiki/Cartesian_closed_categories_containing_Top_and_Prox_as_subcategoriesthis page. Depsite of the fact that this page is written by me, I can’t remember whether that page notation is

compatible

with the notation in my book. Please analyse it and make its copy on PlanetMath.

Title

Background of several open problems I stumbled upon

Canonical name

BackgroundOfSeveralOpenProblemsIStumbledUpon

Date of creation

2013-04-21 12:18:16

Last modified on

2013-04-21 12:18:16

Owner

porton (9363)

Last modified by

porton (9363)

Numerical id

14

Author

porton (9363)

Entry type

Topic

Classification

msc 54J05

Classification

msc 54A05

Classification

msc 54D99

Classification

msc 54E05

Classification

msc 54E17

Classification

msc 54E99

Generated on Sat Feb 10 11:31:48 2018 by

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Conjecture: Every regular paratopological group is Tychonoff

Conjecture: Every regular paratopological group is Tychonoff

A long standing

conjecture

:

¡strong¿Conjecture¡/strong¿ Every

regular

paratopological group is Tychonoff.

I propose to formulate and prove a more general statement about funcoids.

It may may involve product of funcoids, such as cross-composition product or not yet well described

direct product

of continuous funcoids.

Title

Conjecture: Every regular paratopological group is Tychonoff

Canonical

name

ConjectureEveryRegularParatopologicalGroupIsTychonoff

Date of creation

2013-03-31 18:02:01

Last modified on

2013-03-31 18:02:01

Owner

porton (9363)

Last modified by

porton (9363)

Numerical id

1

Author

porton (9363)

Entry type

Definition

Classification

msc 54J05

Classification

msc 54A05

Classification

msc 54D99

Classification

msc 54E05

Classification

msc 54E17

Classification

msc 54E99

Generated on Sat Feb 10 11:31:49 2018 by

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Direct products in a category of funcoids

Direct products in a category of funcoids

ADDED: I’ve proved that subatomic

product

is the categorical product.

There are defined (

\href

http://www.mathematics21.org/algebraic-general-topology.htmlsee my book) several kinds of product of (any possibly

infinite

number) funcoids:

1.

cross-composition product

2.

subatomic product

3.

displaced product

There is one more kind of product, for which it is not proved that the product of funcoids are (pointfree) funcoids:

⟨

f

1

×

f

2

⟩

x

=

⊔

{

⟨

f

1

⟩

X

×

FCD

⟨

f

2

⟩

X

|

X

∈

atoms

x

}

.

It is considered natural by

analogy

with the category

Top

of

topological spaces

to consider this category:

•

Objects are endofuncoids on small sets.

•

Morphisms

between a endofuncoids

μ

and

ν

are

continuous

(that is corresponding to a continuous funcoid)

functions

from the object of

μ

to the object of

ν

.

•

Composition

is

induced

by composition of functions.

It is

trivial

to show that the above is really a category.

The

product of functions

is the same as in

Set

.

Title

Direct products

in a category of funcoids

Canonical

name

DirectProductsInACategoryOfFuncoids

Date of creation

2013-09-12 19:40:35

Last modified on

2013-09-12 19:40:35

Owner

porton (9363)

Last modified by

porton (9363)

Numerical id

11

Author

porton (9363)

Entry type

Definition

Classification

msc 54J05

Classification

msc 54A05

Classification

msc 54D99

Classification

msc 54E05

Classification

msc 54E17

Classification

msc 54E99

Generated on Sat Feb 10 11:31:51 2018 by

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Intersecting two staroidal products

Intersecting two staroidal products

Prerequisites:

\href

http://www.mathematics21.org/algebraic-general-topology.htmlAlgebraic General Topology.

Conjecture.

∏

Strd

a

≭

∏

Strd

b

⇔

b

∈

∏

Strd

a

⇔

a

∈

∏

Strd

b

⇔

∀

i

∈

n

:

a

i

≭

b

i

for every

n

-indexed families

a

and

b

of filters on

powersets

.

The above conjecture has the below

consequence

(see my book for not so long proof).

Conjecture.

Let

f

is a staroid on powersets and

a

∈

∏

i

∈

arity

⁡

f

Src

⁡

f

i

,

b

∈

∏

i

∈

arity

⁡

f

Dst

⁡

f

i

. Then

∏

Strd

a

⁢

[

∏

(

C

)

f

]

⁢

∏

Strd

b

⇔

∀

i

∈

n

:

a

i

⁢

[

f

i

]

⁢

b

i

.

Title

Intersecting two staroidal products

Canonical name

IntersectingTwoStaroidalProducts

Date of creation

2013-03-22 19:50:13

Last modified on

2013-03-22 19:50:13

Owner

porton (9363)

Last modified by

porton (9363)

Numerical id

3

Author

porton (9363)

Entry type

Conjecture

Classification

msc 54J05

Classification

msc 54A05

Classification

msc 54D99

Classification

msc 54E05

Classification

msc 54E17

Classification

msc 54E99

Generated on Sat Feb 10 11:31:53 2018 by

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Intersection with a staroid product through its upper sets

Intersection with a staroid product through its upper sets

Prerequisites:

\href

http://www.mathematics21.org/algebraic-general-topology.htmlAlgebraic General Topology.

Conjecture.

Let

𝔉

is a family of sets of filters on

distributive

lattices with least

elements

. Let

a

∈

∏

𝔉

,

S

∈

𝒫

⁢

∏

𝔉

is a generalized

filter base

,

⊔

S

=

a

,

f

is a staroid of the form

∏

𝔉

. Then

∏

Strd

⁡

(

𝔉

)

a

≭

f

⇔

∀

A

∈

S

:

∏

Strd

⁡

(

𝔄

)

A

≭

f

.

(This conjecture may be weakened for the special case of filters on

powersets

.)

Title

Intersection

with a staroid

product

through its

upper sets

Canonical name

IntersectionWithAStaroidProductThroughItsUpperSets

Date of creation

2013-03-22 19:48:39

Last modified on

2013-03-22 19:48:39

Owner

porton (9363)

Last modified by

porton (9363)

Numerical id

2

Author

porton (9363)

Entry type

Conjecture

Classification

msc 54J05

Classification

msc 54A05

Classification

msc 54D99

Classification

msc 54E05

Classification

msc 54E17

Classification

msc 54E99

Generated on Sat Feb 10 11:31:54 2018 by

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intrinsically knotted

intrinsically knotted

A graph

Γ

is called

intrinsically knotted

if every

embedding

of

Γ

in

ℝ

3

contains a nontrivial knot.

Example:

K

7

, the

complete graph

on 7 vertices, was proven to be intrinsically knotted by Conway and Gordon.

The property of being

not

intrinsically knotted is

inherited by minors

. That is, if a graph

Γ

is not intrinsically knotted and the graph

Γ

′

can be obtained from

Γ

by deleting or contracting edges, then

Γ

′

is also not intrinsically knotted.

According to the

Robertson-Seymour Theorem

(also known as Wagner’s Conjecture), this means that the obstruction set for this property must be finite. Thus there are only a finite number of intrinsically knotted graphs which are minor

minimal

, that is, for which any graph obtained by edge deletion or

contraction

is not intrinsically knotted. As of the creation of this article (06/01/2008), this set is still not known.

Title

intrinsically knotted

Canonical name

IntrinsicallyKnotted

Date of creation

2013-03-22 18:06:08

Last modified on

2013-03-22 18:06:08

Owner

YourInnerNurmo (20577)

Last modified by

YourInnerNurmo (20577)

Numerical id

5

Author

YourInnerNurmo (20577)

Entry type

Definition

Classification

msc 54J05

Related topic

IntrinsicallyLinked

Generated on Sat Feb 10 11:31:56 2018 by

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intrinsically linked

intrinsically linked

A graph

Γ

is

intrinsically linked

if any

embedding

of

Γ

in

ℝ

3

contains a nontrivial link.

Example:

K

6

, the

complete graph

on 6 vertices, was proven to be intrinsically linked by Conway and Gordon.

The

property

of being

not

intrinsically linked is inherited by minors. That is, if

Γ

is not intrinsically linked and

Γ

′

can be obtained from

Γ

by

edge contractions

or deletions, then

Γ

′

is also not intrinsically linked.

By the

Robertson-Seymour Theorem

(formerly Wagner’s

Conjecture

), the obstruction set for this property must be finite. This means that the set of minor

minimal

intrinsically linked graphs is finite. In fact, there are 7 graphs in this set; it is known as the Petersen family, and it consists of graphs which can be obtained from

K

6

by repeated

△

-Y or Y-

△

transformations

.

Title

intrinsically linked

Canonical

name

IntrinsicallyLinked

Date of creation

2013-03-22 18:06:11

Last modified on

2013-03-22 18:06:11

Owner

YourInnerNurmo (20577)

Last modified by

YourInnerNurmo (20577)

Numerical id

7

Author

YourInnerNurmo (20577)

Entry type

Definition

Classification

msc 54J05

Related topic

IntrinsicallyKnotted

Generated on Sat Feb 10 11:31:57 2018 by

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last non-zero digit of factorial

last non-zero digit of factorial

We will show how to compute the last non-zero
digit of the

factorial

of a number from its
digits without having to compute the factorial itself.

Let

L

⁢

(

n

)

denote the last non-zero digit of

n

in base 10.
We note some basic

properties

of

L

which can easily
be checked:

•

For all

n

, we have

L

⁢

(

10

⁢

n

)

=

L

⁢

(

n

)

.

•

If

L

⁢

(

m

)

≠

5

and

L

⁢

(

n

)

≠

5

, then

L

⁢

(

m

⁢

n

)

=

L

⁢

(

L

⁢

(

m

)

⁢

L

⁢

(

n

)

)

.

We also tabulate the values of

L

⁢

(

n

!

)

for small values of

n

:

L

⁢

(

0

!

)

=

1

L

⁢

(

1

!

)

=

1

L

⁢

(

2

!

)

=

2

L

⁢

(

3

!

)

=

6

L

⁢

(

4

!

)

=

4

L

⁢

(

5

!

)

=

2

L

⁢

(

6

!

)

=

2

L

⁢

(

7

!

)

=

4

L

⁢

(

8

!

)

=

2

L

⁢

(

9

!

)

=

8

L

⁢

(

10

!

)

=

8

Next, we make two less trivial observations:

{

theorem

}

For all

positive integers

n

, we have

L

⁢

(

n

!

)

≠

5

.

Title

last non-zero digit of factorial

Canonical

name

LastNonzeroDigitOfFactorial

Date of creation

2013-03-24 0:23:36

Last modified on

2013-03-24 0:23:36

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

4

Author

rspuzio (6075)

Entry type

Definition

Generated on Sat Feb 10 11:31:59 2018 by

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Meta-singular numbers

Meta-singular numbers

Not quite clear exposition and

definition

. Need to be more exact.

In

\href

http://www.mathematics21.org/algebraic-general-topology.htmlmy book I defined (generalized) limit of any (e.g. non-continuous) function.

But applying it, to say, a

differential equation

(replacing the limit in the definition of

derivative

with my generalized limit) does not work because the components of the equation may be of different types (for example generalized limit of a function

ℝ

→

ℝ

is

not

a

real number

).

To overcome this shortcoming I propose what I call

meta-singular numbers

.

[TODO: introduce the term

singularity level above

for a given number system.]

Let

a

is a generalized limit. I will denote

r

⁢

(

a

)

such number (or generalized limit of a lower rank) that

a

=

xlim

⁡

(

{

r

⁢

(

a

)

}

×

𝖥𝖢𝖣

x

)

where

x

is a filter (that is

a

is a limit of a constant function), if such

r

⁢

(

a

)

exists. I will call reduced limit repeated applying

r

⁢

(

…

⁢

r

⁢

(

a

)

⁢

…

)

to a generalized limit

a

.

This definition is for now all I know about meta-singular numbers. It’s yet needed to formulate and prove some

theorems

.

See also

\href

http://www.mathematics21.org/binaries/reduced-limit.pdfthis rough draft on my site (in PDF format).

Title

Meta-singular numbers

Canonical

name

MetasingularNumbers

Date of creation

2013-06-10 20:51:40

Last modified on

2013-06-10 20:51:40

Owner

porton (9363)

Last modified by

porton (9363)

Numerical id

10

Author

porton (9363)

Entry type

Definition

Generated on Sat Feb 10 11:32:01 2018 by

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Multifuncoid has atomic arguments

Multifuncoid has atomic arguments

A counter-example against this conjecture have been found.
See

\href

http://www.mathematics21.org/algebraic-general-topology.htmlAlgebraic General Topology.

Prerequisites:

\href

http://www.mathematics21.org/algebraic-general-topology.htmlAlgebraic General Topology.

Conjecture.

L

∈

[

f

]

⇒

[

f

]

∩

∏

i

∈

dom

⁡

𝔄

atoms

L

i

≠

∅

for every pre-multifuncoid

f

of the form whose

elements

are

atomic posets

.

A weaker conjecture: It is true for forms whose elements are

powersets

.

The following is an attempted (partial) proof:

If

arity

⁡

f

=

0

our

theorem

is

trivial

, so let

arity

⁡

f

≠

0

. Let

⊑

is a

well-ordering

of

arity

⁡

f

with

greatest element

m

.

Let

Φ

is a

function

which maps non-least elements of posets into atoms under these elements and least elements into themselves. (Note that

Φ

is defined on least elements only for completeness,

Φ

is never taken on a least element in the proof below.)

\color

brown [TODO: Fix the ”

universal set

”

paradox

here.]

Define a

transfinite sequence

a

by

transfinite induction

with the

formula

a

c

=

Φ

⁢

⟨

f

⟩

c

⁢

(

a

|

X

⁢

(

c

)

∖

{

c

}

∪

L

|

(

arity

⁡

f

)

∖

X

⁢

(

c

)

)

.

Let

b

c

=

a

|

X

⁢

(

c

)

∖

{

c

}

∪

L

|

(

arity

⁡

f

)

∖

X

⁢

(

c

)

. Then

a

c

=

Φ

⁢

⟨

f

⟩

c

⁢

b

c

.

Let us prove by transfinite induction

a

c

∈

atoms

⁡

L

c

.

a

c

=

Φ

⁢

⟨

f

⟩

c

⁢

L

|

(

arity

⁡

f

)

∖

{

c

}

⊑

⟨

f

⟩

c

⁢

L

|

(

arity

⁡

f

)

∖

{

c

}

. Thus

a

c

⊑

L

c

. [TODO: Is it true for pre-multifuncoids?]

The only thing remained to prove is that

⟨

f

⟩

c

⁢

b

c

≠

0

that is

⟨

f

⟩

c

⁢

(

a

|

X

⁢

(

c

)

∖

{

c

}

∪

L

|

(

arity

⁡

f

)

∖

X

⁢

(

c

)

)

≠

0

that is

y

≭

⟨

f

⟩

c

⁢

b

c

.

Title

Multifuncoid has atomic arguments

Canonical name

MultifuncoidHasAtomicArguments

Date of creation

2014-12-14 21:09:54

Last modified on

2014-12-14 21:09:54

Owner

porton (9363)

Last modified by

porton (9363)

Numerical id

2

Author

porton (9363)

Entry type

Conjecture

Classification

msc 54J05

Classification

msc 54A05

Classification

msc 54D99

Classification

msc 54E05

Classification

msc 54E17

Classification

msc 54E99

Generated on Sat Feb 10 11:32:03 2018 by

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bibliography: doctoral theses supervised by A. Grothendieck

bibliography: doctoral theses supervised by A. Grothendieck

0.1

Doctoral theses supervised by Alexander Grothendieck

http://www.math.jussieu.fr/ leila/grothendieckcircle/theses.pdf

(original list in French)

0.1.1

An incomplete list of “theses directed by A. Grothendieck”:

1.

Schemes of reductive groups, by M. Demazure
[1] Schémas en groupes réductifs; short version of the thesis by M. Demazure, published in

Bull. SMF. Mém

93

(1965), 369-413. (Demazure’s thesis is published “in totto” in SGA 3.

2.

Derived categories

in

l

-adic

cohomology

.
[2] Catégories dérivées en cohomologie l’-adique, (1969), thesis of J.-P.
Jouanolou.

3.

Ample sheaves over

group schemes

and

homogeneous spaces

, thesis of Michel
Raynaud, published in Springer

Lecture Notes

119

(1970).
[3] Faisceaux amples sur les schémas en groupes et espaces homogénes; thesis of Michel
Raynaud.

4.

Lefschetz theorems

in coherent cohomology nad etale cohomology;
[4] Thèorémes de Lefschetz en cohomologie cohérente et en cohomologie étale; thesis of
Michéle Raynaud, Bull. SMF Mém. 41 (1976).

5.

Complexe cotangent et déformations, Springer

Lecture Notes

239

(1971); thesis of Luc
Illusie.

6.

(On)

Tannakian Categories

, Thesis by Saavedra Rivano, published in Springer

Lecture Notes

, 265 (1972).
[6] Catégories Tannakiennes, Springer Lecture Notes 265 (1972); thesis of Saavedra Rivano.

7.

Non-Abelian Cohomology

, Thesis by J. Giraud.
[7] Cohomologie non abélienne,

Grundlehren Math. Wiss.

179 (1972), Springer; thesis
of J. Giraud.

8.

Ring Topos and relative schemes, Thesis by Monique Hakim.
[8] Topos annelés et schémas relatifs,

Ergebnisse Math. Gr.

64

(1972), Springer; thesis
of Monique Hakim.

9.

Cohomologie cristalline, Springer Lecture Notes 407 (1974); thesis of Pierre Berthelot.

10.

Derived Categories and

Abelian Categories

, Thesis by Jean-Louis Verdier.,
[10] Des catégories dérivées des catégories abéliennes; thesis of Jean-Louis Verdier, later
published in

Astérisque

239

(1996), SMF (G. Maltsiniotis, ed.)

11.

Grothendieck Categories

, Thesis of Hong Suan Xsin.
[11] Gr-Categories; thesis of Hong Suan Xsin.

12.

Isotopic

study of

embeddings

of a topological

1

-complex into a complex

surface

; thesis
of Yves Ladegaillerie, short version published in

Topology

23

no. 3(1984), 303-311.

13.

Jacobiennes généralisées globales relatives; thesis of Carlos Contou-Carrére, short version
published in the

Grothendieck Festschrift

, vol. II (1990).

Title

bibliography: doctoral theses supervised by A. Grothendieck

Canonical name

BibliographyDoctoralThesesSupervisedByAGrothendieck

Date of creation

2013-03-22 18:28:15

Last modified on

2013-03-22 18:28:15

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

14

Author

bci1 (20947)

Entry type

Bibliography

Classification

msc 55-00

Classification

msc 18-00

Generated on Sat Feb 10 11:40:23 2018 by

LaTeXML
concepts in abstract algebra

concepts in abstract algebra

The aim of this entry is to present a list of the key

operators

used in

abstract algebra

. Each entry in the list (or will in the future) to the corresponding PlanetMath entry where the object is presented in greater detail. For convenience, this list also presents the encouraged notation to use (at PlanetMath) for these objects.

•

G

group,

subgroup

•

normal subgroup

•

cyclic group

•

group algebra

•

Galois group

•

Polish group

•

G-Set (

http://planetmath.org/G-Set

)

•

groupoid

group

•

𝒢

groupoid

•

semigroup

•

monoid

•

generator

,

generate

•

[

a

,

b

]

commutator

(

http://planetmath.org/DerivedSubgroup

)

•

⟨

g

⟩

cyclic group generated by an

element

•

R

ring,

subring

•

polynomial ring

•

I

,

𝔞

ideal

•

R

/

I

quotient ring

•

S

-

1

⁢

R

localization

of

R

at

S

•

D

integral domain

•

division ring

•

ring group

•

F

,

K

field

•

N

G

⁢

(

H

)

normalizer

of a subgroup

•

C

⁢

(

a

)

centralizer

of an element

•

Z

⁢

(

G

)

center of a group (or centre of a group)

•

H

◁

G

normal subgroup

•

H

⁢

char

⁡

G

characteristic subgroup

•

G

/

H

quotient group

•

⟨

S

G

⟩

normal closure

•

a

⁢

H

,

H

⁢

a

left coset

and

right coset

(

http://planetmath.org/Coset

) respectively

•

element, unit, unity,

inverse

,

identity

•

nilpotent

•

idempotent

•

M

module,

submodule

•

homomorphism

, homomorphy

•

isomorphism

, isomorphy, isomorphic

•

monomorphism

,

epimorphism

•

endomorphism

•

automorphism

General Algebras and Algebroids

•

Universal algebras

•

superalgerbas

•

F-algebras

•

double algebras

•

Quantum Operator algebras

•

general algebras

•

higher dimensional algebras

•

logic algebras

•

quantum logic

algebras

Title

concepts in abstract algebra

Canonical name

ConceptsInAbstractAlgebra

Date of creation

2013-03-22 14:42:38

Last modified on

2013-03-22 14:42:38

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

26

Author

matte (1858)

Entry type

Topic

Classification

msc 55-00

Classification

msc 18-00

Classification

msc 16-00

Classification

msc 13-00

Classification

msc 20-00

Classification

msc 15-00

Synonym

classes of algebras

Synonym

examples of algebras

Generated on Sat Feb 10 11:40:25 2018 by

LaTeXML
example of paracompact topological spaces

example of paracompact topological spaces

0.1

Paracompact space examples

Locally compact Hausdorff spaces

that are also

σ

-

compact

(

http://planetmath.org/AlternativeDefinitionOfSigmaFiniteMeasure

) can be shown to be

paracompact

,
and, therefore, also normal. Other particular cases of

paracompact spaces

are those of locally compact Hausdorff spaces, second countable Hausdorff locally compact spaces, and their related

locally compact groupoids

.

Title

example of paracompact topological spaces

Canonical name

ExampleOfParacompactTopologicalSpaces

Date of creation

2013-03-22 18:23:27

Last modified on

2013-03-22 18:23:27

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

8

Author

bci1 (20947)

Entry type

Example

Classification

msc 55-00

Classification

msc 54-00

Synonym

σ

-compact

Synonym

locally compact Hausdorff spaces

Synonym

second countable Hausdorff locally compact spaces

Related topic

Paracompact

Related topic

LocallyCompactGroupoids

Related topic

LocallyCompactHausdorffSpace

Generated on Sat Feb 10 11:40:27 2018 by

LaTeXML