aarjaneiro/planetmath huggingface dataset:
tubular neighborhood

tubular neighborhood

In mathematics, a

tubular neighborhood

of a

submanifold

of a

smooth manifold

is an

open set

around it resembling the

normal bundle

.

The idea behind a tubular neighborhood can be explained in a simple example. Consider a

smooth curve

in the plane without self-intersections. On each point on the curve draw a line

perpendicular

to the curve. Unless the curve is

straight

, these lines will intersect among themselves in a rather complicated fashion. However, if one looks only in a narrow band around the curve, the portions of the lines in that band will not intersect, and will cover the entire band without gaps. This band is the tubular neighborhood.

In general, let

M

′

be a submanifold of a

manifold

M

, and let

N

be the normal bundle of

M

′

in

M

(

M

′

will play the role of the curve, and

M

will be like the plane containing the curve). Consider the map

i

:

N

0

→

M

′

which establishes a bijective correspondence between the

zero section

N

0

of

N

and the submanifold

M

′

of

M

. The mapping

i

maps the curve (blue in the following

diagram

) at the bottom to the blue curve on top, and each of the infinite lines on the bottom, to each of the finite lines (they can also be curves) on top.

An extension

j

of this map to the entire normal bundle

N

with values in

M

such that

j

⁢

(

N

)

is an open set in

M

and

j

is a

homeomorphism

between

N

and

j

⁢

(

N

)

is called a tubular neighbourhood.

Often times one calls the open set

T

=

j

⁢

(

N

)

, rather than

j

itself, a tubular neighbourhood of

M

′

, it is assumed implicitly that the homeomorphism

j

mapping

N

to

T

exists.

The following schematic illustration of the normal bundle

N

, with the zero section

N

0

in blue. The

transformation

j

maps

N

0

to the curve

M

′

, and

N

to the tubular neighborhood of

M

′

.

References

1

Raoul Bott & Loring W. Tu

Differential forms in algebraic topology

. Berlin: Springer-Verlag. (1982)

2

Waldyr Muniz Oliva

Geometric Mechanics

. Berlin: Springer. (1982)

This entry was adapted from the Wikipedia article

http://en.wikipedia.org/wiki/Tubular_neighborhood

Tubular neighborhood as of June 10, 2007.

These diagrams were created by Oleg Alexandrov and released to the public domain by him.

Title

tubular neighborhood

Canonical name

TubularNeighborhood

Date of creation

2013-03-22 17:13:53

Last modified on

2013-03-22 17:13:53

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

8

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 54A05

Synonym

tubular neighbourhood

Generated on Sat Feb 10 11:13:20 2018 by

LaTeXML
lattice of topologies

lattice of topologies

Let

X

be a set. Let

L

be the set of all

topologies

on

X

. We may order

L

by

inclusion

. When

𝒯

1

⊆

𝒯

2

, we say that

𝒯

2

is finer (

http://planetmath.org/Finer

) than

𝒯

1

, or that

𝒯

2

refines

𝒯

1

.

Theorem 1

.

L

, ordered by inclusion, is a

complete lattice

.

Proof.

Clearly

L

is a

partially ordered set

when ordered by

⊆

. Furthermore, given any family of topologies

𝒯

i

on

X

, their

intersection

⋂

𝒯

i

also defines a topology on

X

. Finally, let

ℬ

i

’s be the corresponding

subbases

for the

𝒯

i

’s and let

ℬ

=

⋃

ℬ

i

. Then

𝒯

generated by

ℬ

is easily seen to be the

supremum

of the

𝒯

i

’s.
∎

Let

L

be the

lattice of topologies

on

X

. Given

𝒯

i

∈

L

,

𝒯

:=

⋁

𝒯

i

is called the

common refinement

of

𝒯

i

. By the proof above, this is the coarsest topology that is than each

𝒯

i

.

If

X

is non-empty with more than one

element

,

L

is also an

atomic lattice

. Each atom is a topology generated by one non-trivial subset of

X

(non-trivial being non-empty and not

X

). The atom has the form

{

∅

,

A

,

X

}

, where

∅

⊂

A

⊂

X

.

Remark

. In general, a

lattice of topologies

on a set

X

is a

sublattice

of

the

lattice of topologies

L

(mentioned above) on

X

.

Title

lattice of topologies

Canonical name

LatticeOfTopologies

Date of creation

2013-03-22 16:54:42

Last modified on

2013-03-22 16:54:42

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 54A10

Related topic

Coarser

Defines

common refinement

Generated on Sat Feb 10 11:13:30 2018 by

LaTeXML
N-Topological Spaces

N-Topological Spaces

N-Topological Spaces

A non-empty set

X

equipped with

N

arbitrary

topologies

𝔍

1

,

𝔍

2

,…..,

𝔍

N

is called an N-topological space and denoted by

(

X

,

𝔍

1

,

…

.

.

,

𝔍

N

)

.

Reference: Khan, K. A., On the possibility of N-topological spaces, International Journal of Mathematical Archive (IJMA) 3 (2012), No.7, 2520-2523

Title

N-Topological Spaces

Canonical

name

NTopologicalSpaces

Date of creation

2013-06-15 11:20:55

Last modified on

2013-06-15 11:20:55

Owner

kamranmetric (1000391)

Last modified by

kamranmetric (1000391)

Numerical id

1

Author

kamranmetric (1000391)

Entry type

Definition

Classification

msc 54A10

Generated on Sat Feb 10 11:13:32 2018 by

LaTeXML
syntopogenous structure

syntopogenous structure

In the early part of the 20th century,

topological spaces

were invented to capture the essence of the idea of continuity. At around the same time, other competing ideas had emerged, resulting in a

variety

of other “

similar

” types of spaces:

uniform spaces

and

proximity spaces

are the two prominent examples. These abstractions have led mathematicians to even further abstractions, in an attempt to combine all these

concepts

into a single construct. One such result is so-called a

syntopogenous structure

.

Before formally defining what a syntopogenous structure is, let us look at some of the commonalities among the three types of spaces that led to this “generalized”

structure

. Specifically, in all three types of sapces, we can define a

transitive relation

on the space such that the

relation

satisfies

some features that are common in all three cases:

Let

X

be a space and

A

,

B

⊆

X

, we define

A

≤

B

iff

•

(topological)

A

⊆

B

∘

, the

interior

of

B

.

•

(uniform)

U

⁢

[

A

]

⊆

B

for some entourage

U

.

U

⁢

[

A

]

is a

uniform neighborhood

of

A

.

•

(proximity)

A

⁢

δ

′

⁢

(

X

-

B

)

, where

δ

is the proximity relation, and

δ

′

is its

complement

.

In all three cases, the relation is

transitive

. Furthermore, we have the following:

1.

∅

≤

∅

,

2.

X

≤

X

,

3.

if

A

≤

B

, then

A

⊆

B

,

4.

if

A

≤

B

and

C

≤

D

, then

A

∩

C

≤

B

∩

D

,

5.

if

A

≤

B

and

C

≤

D

, then

A

∪

C

≤

B

∪

D

,

6.

if

A

⊆

B

≤

C

⊆

D

, then

A

≤

D

.

Definition

. Let

X

be a set. A

topogenous order

≤

on

X

is a

binary relation

on

P

⁢

(

X

)

, the

powerset

of

X

, satisfying the six properties above.

By properties 2 and 6, we see that a topogenous order is a transitive

antisymmetric relation

.

We are now ready for the main definition.

Definition

. A

syntopogenous structure

consists of a set

X

and a

collection

𝒮

of topogenous orders on

X

such that:

•

if

R

1

,

R

2

∈

𝒮

, then there is

R

∈

𝒮

such that

R

1

∩

R

2

⊆

R

,

•

for any

R

∈

𝒮

, then there is

S

∈

𝒮

such that

R

⊆

S

∘

S

.

Remark

. The two defining conditions of a syntopogenous structure

(

X

,

𝒮

)

are

equivalent

to the following, given subsets

A

,

B

of

X

:

•

for any

≤

1

,

≤

2

∈

𝒮

, there is a

≤

∈

𝒮

such that

A

≤

1

B

and

A

≤

2

B

imply

A

≤

B

,

•

for any

≤

1

∈

𝒮

with

A

≤

1

B

, there is a

≤

2

∈

𝒮

such that

A

≤

2

C

≤

2

B

for some subset

C

of

X

.

References

1

A. Császár,

Foundations

of General Topology

,
Macmillan, New York, 1963.

2

S.A. Naimpally, B.D. Warrack,

Proximity Spaces

, Cambridge University Press, 1970.

Title

syntopogenous structure

Canonical name

SyntopogenousStructure

Date of creation

2013-03-22 16:57:40

Last modified on

2013-03-22 16:57:40

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 54A15

Defines

topogenous order

Generated on Sat Feb 10 11:13:41 2018 by

LaTeXML
accumulation points and convergent subnets

accumulation points and convergent subnets

Proposition

.

Let

X

be a

topological space

and

(

x

α

)

α

∈

A

a net in

X

. A point

x

∈

X

is an

accumulation point

of

(

x

α

)

if and only if some subnet of

(

x

α

)

converges

to

x

.

Proof.

Suppose first that

(

x

α

β

)

β

∈

B

is a subnet of

(

x

α

)

converging to

x

. Given an

open subset

U

of

X

containing

x

and

α

∈

A

, we may select

β

1

∈

B

such that

x

α

β

∈

U

for

β

≥

β

1

, as well as

β

2

∈

B

such that

α

β

≥

α

for

β

≥

β

2

. Finally, because

B

is directed, there exists

β

∈

B

such that

β

≥

β

1

and

β

≥

β

2

; we then have

α

β

≥

α

and

x

α

β

∈

U

, so that

(

x

α

)

is

frequently in

U

, whence

x

is an accumulation point of

(

x

α

)

.

Conversely

, suppose that

x

is an accumulation point of

(

x

α

)

, let

N

be the set of open neighborhoods of

x

in

X

, directed by reverse

inclusion

, and let

B

=

A

×

N

, directed in the natural way. For each pair

(

γ

,

U

)

∈

B

, select

α

(

γ

,

U

)

∈

B

such that

α

≥

γ

and

x

α

(

γ

,

U

)

∈

U

;

(

x

α

(

γ

,

U

)

)

(

γ

,

U

)

∈

B

is then a subnet of

(

x

α

)

that converges to

x

, for given

U

∈

N

and

γ

∈

A

, if

(

γ

′

,

U

′

)

≥

(

γ

,

U

)

, then

α

(

γ

′

,

U

)

≥

γ

′

≥

γ

and

x

α

(

γ

′

,

U

′

)

∈

U

′

⊆

U

.
∎

Title

accumulation points and convergent subnets

Canonical name

AccumulationPointsAndConvergentSubnets

Date of creation

2013-03-22 18:37:40

Last modified on

2013-03-22 18:37:40

Owner

azdbacks4234 (14155)

Last modified by

azdbacks4234 (14155)

Numerical id

9

Author

azdbacks4234 (14155)

Entry type

Theorem

Classification

msc 54A20

Related topic

Net

Related topic

Neighborhood

Related topic

DirectedSet

Related topic

CompactnessAndConvergentSubnets

Generated on Sat Feb 10 11:13:44 2018 by

LaTeXML
alternative characterization of ultrafilter

alternative characterization of ultrafilter

Let

X

be a set.
A filter

ℱ

over

X

is an

ultrafilter

if and only if
it

satisfies

the following condition:
if

A

⁢

∐

B

=

X

(see

disjoint union

),
then either

A

∈

ℱ

or

B

∈

ℱ

.

This result can be generalized somewhat:
a filter

ℱ

over

X

is an ultrafilter if and only if
it satisfies the following condition:
if

A

∪

B

=

X

(see union), then either

A

∈

ℱ

or

B

∈

ℱ

.

This

theorem

can be extended to
the following two

propositions

about finite unions:

1.

A filter

ℱ

over

X

is an ultrafilter if and only if,
whenever

A

1

,

…

,

A

n

are subsets of

X

such that

∐

i

=

1

n

A

i

=

X

then there exists exactly one

i

such that

A

i

∈

ℱ

.

2.

A filter

ℱ

over

X

is an ultrafilter if and only if,
whenever

A

1

,

…

,

A

n

are subsets of

X

such that

⋃

i

=

1

n

A

i

=

X

then there exists an

i

such that

A

i

∈

ℱ

.

Title

alternative characterization of ultrafilter

Canonical name

AlternativeCharacterizationOfUltrafilter

Date of creation

2013-03-22 14:42:20

Last modified on

2013-03-22 14:42:20

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

13

Author

yark (2760)

Entry type

Theorem

Classification

msc 54A20

Generated on Sat Feb 10 11:13:46 2018 by

LaTeXML
Banach fixed point theorem

Banach fixed point theorem

Let

(

X

,

d

)

be a

complete metric space

. A function

T

:

X

→

X

is said to be a

contraction mapping

if there is a constant

q

with

0

≤

q

<

1

such that

d

⁢

(

T

⁢

x

,

T

⁢

y

)

≤

q

⋅

d

⁢

(

x

,

y

)

for all

x

,

y

∈

X

. Contractions have an important property.

Theorem 1

(Banach Theorem).

Every contraction has a unique

http://planetmath.org/node/2777

fixed point

.

There is an estimate to this fixed point that can be useful in applications. Let

T

be a contraction mapping on

(

X

,

d

)

with constant

q

and unique fixed point

x

*

∈

X

. For any

x

0

∈

X

, define recursively the following

sequence

x

1

:=

T

⁢

x

0

x

2

:=

T

⁢

x

1

⋮

x

n

+

1

:=

T

⁢

x

n

.

The following

inequality

then holds:

d

⁢

(

x

*

,

x

n

)

≤

q

n

1

-

q

⁢

d

⁢

(

x

1

,

x

0

)

.

So the sequence

(

x

n

)

converges

to

x

*

. This estimate is occasionally responsible for this result being known as

the method of successive approximations

.

Title

Banach fixed point theorem

Canonical name

BanachFixedPointTheorem

Date of creation

2013-03-22 12:31:10

Last modified on

2013-03-22 12:31:10

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

21

Author

mathwizard (128)

Entry type

Theorem

Classification

msc 54A20

Classification

msc 47H10

Classification

msc 54H25

Synonym

contraction principle

Synonym

contraction mapping theorem

Synonym

method of successive approximations

Synonym

Banach-Caccioppoli fixed point theorem

Related topic

FixedPoint

Defines

contraction mapping

Defines

contraction operator

Generated on Sat Feb 10 11:13:53 2018 by

LaTeXML
compactness and accumulation points of nets

compactness and accumulation points of nets

Theorem

.

A

topological space

X

is

compact

if and only if every net in

X

has an

accumulation point

.

Proof.

Suppose

X

is compact and let

(

x

α

)

α

∈

A

be a net in

X

. For each

α

∈

A

, put

E

α

=

{

x

β

:

β

≥

α

}

; the

collection

{

E

α

¯

:

α

∈

A

}

of

closed subsets

of

X

has the

finite intersection property

, for given

α

1

,

…

,

α

n

∈

A

, because

A

is directed, there exists

β

∈

A

satisfying

β

≥

α

i

for each

i

∈

{

1

,

…

,

n

}

, so that

x

β

∈

⋂

i

=

1

n

E

α

i

¯

. Therefore, by

compactness

,

⋂

α

∈

A

E

α

¯

≠

∅

; let

x

be a point of this

intersection

. If

U

is any

open subset

of

X

and

α

∈

A

, then because

x

∈

E

α

¯

,

E

α

∩

U

≠

∅

, and thus there exists

β

≥

α

∈

A

for which

x

β

∈

U

. It follows that

x

is an accumulation point of

(

x

α

)

. For the

converse

, assume that

X

fails to be compact, and let

{

U

i

:

i

∈

I

}

be an

open cover

of

X

with no finite subcover. If

B

is the set of finite subsets of

I

, then

B

is directed by

inclusion

. For each set

S

∈

B

, let

x

S

be a point in the

complement

of

⋃

i

∈

S

U

i

. We contend that the net

(

x

S

)

S

∈

B

has no accumulation points; indeed, given

x

∈

X

, we may select

i

0

∈

I

such that

x

∈

U

i

0

; if

S

∈

B

is such that

i

0

∈

S

, that is, if

S

≥

{

i

0

}

, then by construction,

x

S

∉

U

i

0

, establishing our contention.
∎

Corollary

.

The following conditions on a topological space

X

are

equivalent

:

1.

X

is compact;

2.

every net in

X

has an accumulation point;

3.

every net in

X

has a

convergent

subnet;

Proof.

The preceding

theorem

establishes the equivalence of (1) and (2), while that of (2) and (3) is established in the entry on

accumulation points and convergent subnets

.
∎

Title

compactness and accumulation points of nets

Canonical name

CompactnessAndAccumulationPointsOfNets

Date of creation

2013-03-22 18:37:50

Last modified on

2013-03-22 18:37:50

Owner

azdbacks4234 (14155)

Last modified by

azdbacks4234 (14155)

Numerical id

7

Author

azdbacks4234 (14155)

Entry type

Theorem

Classification

msc 54A20

Related topic

Net

Related topic

Compact

Related topic

AccumulationPointsAndConvergentSubnets

Generated on Sat Feb 10 11:13:56 2018 by

LaTeXML
continuity and convergent nets

continuity and convergent nets

Theorem

.

Let

X

and

Y

be

topological spaces

. A

function

f

:

X

→

Y

is

continuous at

a point

x

∈

X

if and only if for each net

(

x

α

)

α

∈

A

in

X

converging to

x

, the net

(

f

⁢

(

x

α

)

)

α

∈

A

converges

to

f

⁢

(

x

)

.

Proof.

If

f

is

continuous

,

(

x

α

)

α

∈

A

converges to

x

, and

V

is an open neighborhood of

f

⁢

(

x

)

in

Y

, then

f

-

1

⁢

(

V

)

is an open neighborhood of

x

in

X

, so there exists

α

0

∈

A

such that

x

α

∈

f

-

1

⁢

(

V

)

for

α

≥

α

0

. It follows that

f

⁢

(

x

α

)

∈

V

for

α

≥

α

0

, hence that

f

⁢

(

x

α

)

→

f

⁢

(

x

)

.

Conversely

, suppose there exists a net

(

x

α

)

α

∈

A

in

X

converging to

x

such that

(

f

⁢

(

x

α

)

)

α

∈

A

does not converge to

f

⁢

(

x

)

, so that, for some

open subset

V

of

Y

containing

f

⁢

(

x

)

and every

α

0

∈

A

, there exists

α

≥

α

0

∈

A

such that

f

⁢

(

x

α

)

∉

V

, hence such that

x

α

∉

f

-

1

⁢

(

V

)

; as

x

α

→

x

by

hypothesis

, this implies that

f

-

1

⁢

(

V

)

cannot be a

neighborhood

of

x

, and thus that

f

fails to be continuous at

x

.
∎

Title

continuity and convergent nets

Canonical name

ContinuityAndConvergentNets

Date of creation

2013-03-22 18:37:53

Last modified on

2013-03-22 18:37:53

Owner

azdbacks4234 (14155)

Last modified by

azdbacks4234 (14155)

Numerical id

5

Author

azdbacks4234 (14155)

Entry type

Theorem

Classification

msc 54A20

Related topic

Net

Related topic

Continuous

Generated on Sat Feb 10 11:13:58 2018 by

LaTeXML
continuous convergence

continuous convergence

Let

(

X

,

d

)

and

(

Y

,

ρ

)

be

metric spaces

, and let

f

n

:

X

⟶

Y

be a sequence of functions. We say that

f

n

converges continuously

to

f

at

x

if

f

n

⁢

(

x

n

)

⟶

f

⁢

(

x

)

for every sequence

(

x

n

)

n

⊂

X

such that

x

n

⟶

x

∈

X

. We say that

f

n

converges continuously

to

f

if it does for every

x

∈

X

.

Title

continuous convergence

Canonical

name

ContinuousConvergence

Date of creation

2013-03-22 14:04:58

Last modified on

2013-03-22 14:04:58

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

7

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 54A20

Synonym

converges continuously

Generated on Sat Feb 10 11:14:00 2018 by

LaTeXML
contractive maps are uniformly continuous

contractive maps are uniformly continuous

Theorem

A

contraction mapping

is

uniformly continuous

.

Proof

Let

T

:

X

→

X

be a contraction mapping in a

metric space

X

with metric

d

. Thus, for some

q

∈

[

0

,

1

)

, we have
for all

x

,

y

∈

X

,

d

⁢

(

T

⁢

x

,

T

⁢

y

)

≤

q

⁢

d

⁢

(

x

,

y

)

.

To prove that

T

is uniformly continuous, let

ε

>

0

be given.
There are two cases.
If

q

=

0

, our claim is

trivial

, since then for all

x

,

y

∈

X

,

d

⁢

(

T

⁢

x

,

T

⁢

y

)

=

0

<

ε

.

On the other hand, suppose

q

∈

(

0

,

1

)

. Then for all

x

,

y

∈

X

with

d

⁢

(

x

,

y

)

<

ε

/

q

, we have

d

⁢

(

T

⁢

x

,

T

⁢

y

)

≤

q

⁢

d

⁢

(

x

,

y

)

<

ε

.

In

conclusion

,

T

is uniformly continuous.

□

The result is stated without proof in

[

1

]

, pp. 221.

References

1

W. Rudin,

Principles of

Mathematical Analysis

, McGraw-Hill Inc., 1976.

Title

contractive maps are uniformly continuous

Canonical name

ContractiveMapsAreUniformlyContinuous

Date of creation

2013-03-22 13:46:28

Last modified on

2013-03-22 13:46:28

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 54A20

Generated on Sat Feb 10 11:14:02 2018 by

LaTeXML
Dini’s theorem

Dini’s theorem

If a

monotonically increasing

net

{

f

n

}

of

continuous

real-valued functions on a

topological space

(

X

,

τ

)

converges

pointwise

to a

continuous function

f

, then the net converges to

f

uniformly on

compacts

.

Title

Dini’s

theorem

Canonical name

DinisTheorem

Date of creation

2013-03-22 12:20:32

Last modified on

2013-03-22 12:20:32

Owner

drini (3)

Last modified by

drini (3)

Numerical id

4

Author

drini (3)

Entry type

Theorem

Classification

msc 54A20

Related topic

TopologicalSpace

Related topic

Compact

Generated on Sat Feb 10 11:14:04 2018 by

LaTeXML
every filter is contained in an ultrafilter

every filter is contained in an ultrafilter

Let

X

be a set and

ℱ

be a filter on

X

. Then there exists an

ultrafilter

𝒰

on

X

which is finer than

ℱ

.

An importance consequnce of this

theorem

is the

existence

of free ultrafilters on

infinite sets

. According to the theorem, there must exist an ultrafilter which is finer than the

cofinite filter

. Since the cofinite filter is free, every filter finer than it must also be free, and hence there exists a free ultafilter.

Also note that this theorem requires the

axiom of choice

.

Title

every filter is

contained

in an ultrafilter

Canonical name

EveryFilterIsContainedInAnUltrafilter

Date of creation

2013-03-22 14:41:41

Last modified on

2013-03-22 14:41:41

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

6

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 54A20

Related topic

LindenbaumsLemma

Generated on Sat Feb 10 11:14:05 2018 by

LaTeXML
every net has a universal subnet

every net has a universal subnet

Theorem

- (Kelley’s theorem) -

Let

X

be a non-empty set. Every net

(

x

α

)

α

∈

𝒜

in

X

has a

universal

subnet (

http://planetmath.org/

Ultranet

). That is, there is a subnet such that for every

E

⊆

X

either the subnet is

eventually

in

E

or eventually in

X

-

E

.

Proof :

Let

ℱ

be a

section filter

for the net

(

x

α

)

α

∈

𝒜

.

Let

𝒟

=

{

(

α

,

U

)

:

α

∈

𝒜

,

U

∈

ℱ

,

x

α

∈

U

}

.

𝒟

is a

directed set

under the order relation given by

(

α

,

U

)

≤

(

β

,

V

)

⟺

{

α

≤

β

V

⊆

U

The map

f

:

𝒟

⟶

𝒜

defined by

f

⁢

(

α

,

U

)

:=

α

is order preserving and

cofinal

. Therefore there is a subnet

(

y

(

α

,

U

)

)

(

α

,

U

)

∈

𝒟

of

(

x

α

)

α

∈

𝒜

associated with the map

f

(that is,

y

(

α

,

U

)

=

x

α

).

We now prove that

(

y

(

α

,

U

)

)

(

α

,

U

)

∈

𝒟

is a net.

Let

E

⊆

X

. We have that

(

y

(

α

,

U

)

)

(

α

,

U

)

∈

𝒟

is

frequently in

E

or frequently in

X

-

E

.

Suppose

(

y

(

α

,

U

)

)

(

α

,

U

)

∈

𝒟

is frequently in

E

.

Let

A

∈

ℱ

and

S

⁢

(

α

)

:=

{

x

β

:

α

≤

β

}

. We have that

S

⁢

(

α

)

∈

ℱ

by definition of section filter.

As

ℱ

is a filter,

A

∩

S

⁢

(

α

)

≠

∅

and so there exists

β

with

α

≤

β

such that

x

β

∈

A

. Hence,

(

β

,

A

)

∈

𝒟

.

As

(

y

(

α

,

U

)

)

(

α

,

U

)

∈

𝒟

is frequently in

E

, there exists

(

γ

,

B

)

∈

𝒟

with

(

β

,

A

)

≤

(

γ

,

B

)

such that

y

(

γ

,

B

)

∈

E

.

Also,

y

(

γ

,

B

)

is in

B

, and therefore, in

A

. So

A

∩

E

≠

∅

.

We conclude that

E

∩

A

≠

∅

for every

A

∈

ℱ

. Therefore,

ℱ

∪

{

E

}

a filter in

X

. As

ℱ

is a

maximal filter

we conclude that

E

∈

ℱ

, and consequently,

(

γ

,

E

)

∈

𝒟

.

We can now see that for every

(

δ

,

C

)

with

(

γ

,

E

)

≤

(

δ

,

C

)

,

y

(

δ

,

C

)

is in

C

and so is in

E

. Therefore,

(

y

(

α

,

U

)

)

(

α

,

U

)

∈

𝒟

is eventually in

E

.

Remark:

If

(

y

(

α

,

U

)

)

(

α

,

U

)

∈

𝒟

is frequently in

X

-

E

, by an analogous we can conclude that it is eventually in

X

-

E

.

This proves that

(

y

(

α

,

U

)

)

(

α

,

U

)

∈

𝒟

is a subnet of

(

x

α

)

α

∈

𝒜

.

□

Title

every net has a universal subnet

Canonical name

EveryNetHasAUniversalSubnet

Date of creation

2013-03-22 17:25:16

Last modified on

2013-03-22 17:25:16

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

8

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 54A20

Synonym

Kelley’s theorem

Related topic

Ultranet

Generated on Sat Feb 10 11:14:08 2018 by

LaTeXML