aarjaneiro/planetmath huggingface dataset:
net

net

Let

X

be a set. A

net

is a map from a directed set to

X

.
In other words, it is a pair

(

A

,

γ

)

where

A

is a directed set and

γ

is a map from

A

to

X

.
If

a

∈

A

then

γ

⁢

(

a

)

is normally written

x

a

,
and then the net is written

(

x

a

)

a

∈

A

,
or simply

(

x

a

)

if the direct set

A

is understood.

Now suppose

X

is a

topological space

,

A

is a directed set,
and

(

x

a

)

a

∈

A

is a net. Let

x

∈

X

.
Then

(

x

a

)

is said to

converge

to

x

if
whenever

U

is an open

neighbourhood

of

x

,
there is some

b

∈

A

such that

x

a

∈

U

whenever

a

≥

b

.

Similarly,

x

is said to be an

accumulation point

(or

cluster point

)
of

(

x

a

)

if whenever

U

is an open neighbourhood of

x

and

b

∈

A

there is

a

∈

A

such that

a

≥

b

and

x

a

∈

U

.

Nets are sometimes called

Moore–Smith

sequences

,
in which case convergence of nets may be called

Moore–Smith convergence

.

If

B

is another directed set,
and

δ

:

B

→

A

is an

increasing

map
such that

δ

⁢

(

B

)

is cofinal in

A

,
then the pair

(

B

,

γ

∘

δ

)

is said to be a

subnet

of

(

A

,

γ

)

.
Alternatively, a subnet of a net

(

x

α

)

α

∈

A

is sometimes defined to be a net

(

x

α

β

)

β

∈

B

such that for each

α

0

∈

A

there exists a

β

0

∈

B

such that

α

β

≥

α

0

for all

β

≥

β

0

.

Nets are a generalisation of sequences (

http://planetmath.org/Sequence

),
and in many respects they work better in arbitrary topological spaces
than sequences do. For example:

•

If

X

is

Hausdorff

then any net in

X

converges to at most one point.

•

If

Y

is a

subspace

of

X

then

x

∈

Y

¯

if and only if there is a net in

Y

converging to

x

.

•

if

X

′

is another topological space
and

f

:

X

→

X

′

is a map,
then

f

is

continuous at

x

if and only if
whenever

(

x

a

)

is a net converging to

x

,

(

f

⁢

(

x

a

)

)

is a net converging to

f

⁢

(

x

)

.

•

X

is

compact

if and only if every net has a

convergent

subnet.

Title

net

Canonical name

Net

Date of creation

2013-03-22 12:54:03

Last modified on

2013-03-22 12:54:03

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

12

Author

yark (2760)

Entry type

Definition

Classification

msc 54A20

Synonym

Moore-Smith sequence

Related topic

Filter

Related topic

NetsAndClosuresOfSubspaces

Related topic

ContinuityAndConvergentNets

Related topic

CompactnessAndConvergentSubnets

Related topic

AccumulationPointsAndConvergentSubnets

Related topic

TestingForContinuityViaNets

Defines

subnet

Defines

Moore-Smith convergence

Defines

cluster point

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

LaTeXML
nets and closures of subspaces

nets and closures of subspaces

Theorem

.

A point of a

topological space

is in the

closure

of a

subspace

if and only if there is a net of points of the subspace converging to the point.

Proof.

Let

X

be a topological space,

x

a point of

X

, and

A

a subspace of

X

.
Suppose first that

x

∈

A

¯

, and let

𝒰

be the

collection

of

neighborhoods

of

x

,

http://planetmath.org/node/123

partially ordered by reverse . For each

U

∈

𝒰

, select a point

x

U

∈

U

∩

A

(such a point is guaranteed to exist because

x

∈

A

¯

); then

(

x

U

)

U

∈

𝒰

is a net of points in

A

, and we claim that

x

U

→

x

. To see this, let

V

be a neighborhood of

x

in

X

, and note that, by construction,

x

V

∈

V

; furthermore, if

U

∈

𝒰

satisfies

V

⊃

U

, then because

x

U

∈

U

,

x

U

∈

V

. It follows that

x

U

→

x

.

Conversely

, suppose there exists a net

(

x

α

)

α

∈

J

of points of

A

converging to

x

, and let

U

⊂

X

be a neighborhood of

x

. Since

x

α

→

x

, there exists

β

∈

J

such that

x

α

∈

U

whenever

β

⪯

α

. Because

x

α

∈

A

for each

α

∈

J

by

hypothesis

, we may conclude that

U

∩

A

≠

∅

, hence that

x

∈

A

¯

.
∎

The forward

implication

of the preceding is a

generalization

of the result that a point of a topological space is in the closure of a subspace if there is a

sequence

of points of the subspace converging to the point, as a sequence is just a net with the positive integers as its

http://planetmath.org/node/360

domain; however, the

converse

(if a point is in the closure of a subspace then there exists a sequence of points of the subspace converging to the point) requires the additional condition that the ambient topological space be

first countable

.

Title

nets and closures of subspaces

Canonical

name

NetsAndClosuresOfSubspaces

Date of creation

2013-03-22 17:18:34

Last modified on

2013-03-22 17:18:34

Owner

azdbacks4234 (14155)

Last modified by

azdbacks4234 (14155)

Numerical id

11

Author

azdbacks4234 (14155)

Entry type

Theorem

Classification

msc 54A20

Related topic

Net

Related topic

DirectedSet

Related topic

PartialOrder

Generated on Sat Feb 10 11:14:13 2018 by

LaTeXML
proof of Banach fixed point theorem

proof of Banach fixed point theorem

Let

(

X

,

d

)

be a non-empty, complete metric space, and let

T

be a

contraction mapping

on

(

X

,

d

)

with

constant

q

. Pick an arbitrary

x

0

∈

X

, and define the

sequence

(

x

n

)

n

=

0

∞

by

x

n

:=

T

n

⁢

x

0

. Let

a

:=

d

⁢

(

T

⁢

x

0

,

x

0

)

. We first show by

induction

that
for any

n

≥

0

,

d

⁢

(

T

n

⁢

x

0

,

x

0

)

≤

1

-

q

n

1

-

q

⁢

a

.

For

n

=

0

, this is obvious. For any

n

≥

1

, suppose that

d

⁢

(

T

n

-

1

⁢

x

0

,

x

0

)

≤

1

-

q

n

-

1

1

-

q

⁢

a

. Then

d

⁢

(

T

n

⁢

x

0

,

x

0

)

≤

d

⁢

(

T

n

⁢

x

0

,

T

n

-

1

⁢

x

0

)

+

d

⁢

(

x

0

,

T

n

-

1

⁢

x

0

)

≤

q

n

-

1

⁢

d

⁢

(

T

⁢

x

0

,

x

0

)

+

1

-

q

n

-

1

1

-

q

⁢

a

=

q

n

-

1

-

q

n

1

-

q

⁢

a

+

1

-

q

n

-

1

1

-

q

⁢

a

=

1

-

q

n

1

-

q

⁢

a

by the

triangle inequality

and repeated application of the property

d

⁢

(

T

⁢

x

,

T

⁢

y

)

≤

q

⁢

d

⁢

(

x

,

y

)

of

T

. By induction, the

inequality

holds for
all

n

≥

0

.

Given any

ϵ

>

0

, it is possible to choose a

natural number

N

such that

q

n

1

-

q

⁢

a

<

ϵ

for all

n

≥

N

, because

q

n

1

-

q

⁢

a

→

0

as

n

→

∞

. Now, for any

m

,

n

≥

N

(we
may assume that

m

≥

n

),

d

⁢

(

x

m

,

x

n

)

=

d

⁢

(

T

m

⁢

x

0

,

T

n

⁢

x

0

)

≤

q

n

⁢

d

⁢

(

T

m

-

n

⁢

x

0

,

x

0

)

≤

q

n

⁢

1

-

q

m

-

n

1

-

q

⁢

a

<

q

n

1

-

q

⁢

a

<

ϵ

,

so the sequence

(

x

n

)

is a

Cauchy sequence

. Because

(

X

,

d

)

is

complete

, this implies that the sequence has a limit in

(

X

,

d

)

;
define

x

*

to be this limit. We now prove that

x

*

is a

fixed
point

of

T

. Suppose it is not, then

δ

:=

d

⁢

(

T

⁢

x

*

,

x

*

)

>

0

.
However, because

(

x

n

)

converges

to

x

*

, there is a natural
number

N

such that

d

⁢

(

x

n

,

x

*

)

<

δ

/

2

for all

n

≥

N

. Then

d

⁢

(

T

⁢

x

*

,

x

*

)

≤

d

⁢

(

T

⁢

x

*

,

x

N

+

1

)

+

d

⁢

(

x

*

,

x

N

+

1

)

≤

q

⁢

d

⁢

(

x

*

,

x

N

)

+

d

⁢

(

x

*

,

x

N

+

1

)

<

δ

/

2

+

δ

/

2

=

δ

,

contradiction

. So

x

*

is a fixed point of

T

. It is also unique.
Suppose there is another fixed point

x

′

of

T

; because

x

′

≠

x

*

,

d

⁢

(

x

′

,

x

*

)

>

0

. But then

d

⁢

(

x

′

,

x

*

)

=

d

⁢

(

T

⁢

x

′

,

T

⁢

x

*

)

≤

q

⁢

d

⁢

(

x

′

,

x

*

)

<

d

⁢

(

x

′

,

x

*

)

,

contradiction. Therefore,

x

*

is the unique fixed point of

T

.

Title

proof of

Banach fixed point theorem

Canonical name

ProofOfBanachFixedPointTheorem

Date of creation

2013-03-22 13:08:34

Last modified on

2013-03-22 13:08:34

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

5

Author

asteroid (17536)

Entry type

Proof

Classification

msc 54A20

Classification

msc 47H10

Classification

msc 54H25

Generated on Sat Feb 10 11:14:20 2018 by

LaTeXML
proof of Dini’s theorem

proof of Dini’s theorem

Without loss of generality we will assume that

X

is

compact

and, by replacing

f

n

with

f

-

f

n

, that the net

converges

monotonically

to 0.

Let

ϵ

>

0

.
For each

x

∈

X

, we can choose an

n

x

, such that

f

n

x

⁢

(

x

)

<

ϵ

/

2

. Since

f

n

x

is

continuous

,
there is an open

neighbourhood

U

x

of

x

, such that for each

y

∈

U

x

, we have

f

n

x

⁢

(

y

)

<

ϵ

/

2

. The

open sets

U

x

cover

X

, which is compact, so we can choose
finitely many

x

1

,

…

,

x

k

such that the

U

x

i

also cover

X

. Then,
if

N

≥

n

x

1

,

…

,

n

x

k

, we have

f

n

⁢

(

x

)

<

ϵ

for each

n

≥

N

and

x

∈

X

, since the

sequence

f

n

is

monotonically decreasing

.
Thus,

{

f

n

}

converges to 0 uniformly on

X

, which was to be proven.

Title

proof of Dini’s

theorem

Canonical name

ProofOfDinisTheorem

Date of creation

2013-03-22 12:44:13

Last modified on

2013-03-22 12:44:13

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

5

Author

mathcam (2727)

Entry type

Proof

Classification

msc 54A20

Generated on Sat Feb 10 11:14:22 2018 by

LaTeXML
proof of every filter is contained in an ultrafilter (alternate proof)

proof of every filter is contained in an ultrafilter (alternate proof)

Let

𝔘

be the family of filters over

X

which are finer than

ℱ

, under the

partial order

of

inclusion

.

Claim 1.

Every chain in

U

has an

upper bound

also in

U

.

Proof.

Take any chain

ℭ

in

𝔘

, and consider the set

𝒞

=

∪

ℭ

. Then

𝒞

is also a filter: it cannot contain the

empty set

, since no filter in the chain does; the

intersection

of two sets in

𝒞

must be present in the filters of

ℭ

; and

𝒞

is

closed under

supersets

because every
filter in

ℭ

is. Obviously

𝒞

is finer than

ℱ

.
∎

So we conclude, by Zorn’s lemma, that

𝔘

must have a

maximal filter

say

𝒰

, which must contain

ℱ

. All we need to show is that

𝒰

is an

ultrafilter

. Now, for any filter

𝒰

, and any set

Y

⊆

X

, we must have:

Claim 2.

Either

U

1

=

{

Z

∩

Y

:

Z

∈

U

}

or

U

2

=

{

Z

∩

(

X

\

Y

)

:

Z

∈

U

}

(or both) are a

filter subbasis

.

Proof.

We prove by

contradiction

that at least one of

𝒰

1

or

𝒰

2

must have the

finite intersection property

. If neither has the finite intersection property, then for some

Z

1

,

…

⁢

Z

k

we must have

∅

=

⋂

1

≤

i

≤

k

Z

i

∩

Y

=

⋂

1

≤

i

≤

k

Z

i

∩

(

X

\

Y

)

.

But then

∅

=

(

⋂

1

≤

i

≤

k

Z

i

∩

Y

)

∪

(

⋂

1

≤

i

≤

k

Z

i

∩

(

X

\

Y

)

)

=

⋂

1

≤

i

≤

k

Z

i

,

and so

𝒰

does not have the finite intersection property either. This cannot be, since

𝒰

is a filter.
∎

Now, by Claim 2, if

𝒰

were not an ultrafilter, i.e., if for some

Y

subset of

X

we would have neither

Y

nor

X

\

Y

in

𝒰

, then the filter generated

𝒰

1

or

𝒰

2

would be finer than

𝒰

, and then

𝒰

would not be maximal.

So

𝒰

is an ultrafilter containing

ℱ

, as intended.

Title

proof of every filter is contained in an ultrafilter (alternate proof)

Canonical name

ProofOfEveryFilterIsContainedInAnUltrafilteralternateProof

Date of creation

2013-03-22 17:52:54

Last modified on

2013-03-22 17:52:54

Owner

brunoloff (19748)

Last modified by

brunoloff (19748)

Numerical id

8

Author

brunoloff (19748)

Entry type

Proof

Classification

msc 54A20

Generated on Sat Feb 10 11:14:24 2018 by

LaTeXML
proof that every filter is contained in an ultrafilter

proof that every filter is contained in an ultrafilter

Let

Y

be the set of all non-empty subsets of

X

which are not

contained

in

ℱ

. By Zermelo’s well-orderding

theorem

, there exists a

relation

‘

≻

’ which well-orders

Y

. Define

Y

′

=

{

0

}

∪

Y

and extend the relation ‘

≻

’ to

Y

′

by decreeing that

0

≺

y

for all

y

∈

Y

.

We shall construct a family of filters

S

i

indexed by

Y

′

using

transfinite induction

. First, set

S

0

=

ℱ

. Next, suppose that, for some

j

∈

Y

,

S

i

has already been defined when

i

≺

j

. Consider the set

⋃

i

≺

j

S

i

; if

A

and

B

are elements of this set, there must exist an

i

≺

j

such that

A

∈

S

i

and

B

∈

S

i

; hence,

A

∩

B

cannot be empty. If, for some

i

≺

j

there exists an

element

f

∈

S

i

such that

f

∩

j

is empty, let

S

j

be the

filter generated by

the

filter subbasis

⋃

i

≺

j

S

i

. Otherwise

{

j

}

∪

⋃

i

≺

j

S

i

is a filter subbasis; let

S

j

be the filter it generates.

Note that, by this

definition

, whenever

i

≺

j

, it follows that

S

i

⊆

S

j

; in particular, for all

i

∈

Y

′

we have

ℱ

⊆

S

i

. Let

𝒰

=

⋃

i

≺

j

S

i

. It is clear that

∅

∉

ℱ

and that

ℱ

⊆

𝒰

.

It is easy to see that

𝒰

is a filter. Suppose that

A

∩

B

∈

𝒰

. Then there must exist an

i

∈

Y

′

such that

A

∩

B

∈

S

i

. Since

S

i

is a filter,

A

∈

S

i

and

B

∈

S

i

, hence

A

∈

𝒰

and

B

∈

𝒰

.

Conversely

, if

A

∈

𝒰

and

B

∈

𝒰

, then there exists an

i

∈

Y

′

such that

A

∈

S

i

and

B

∈

S

i

. Since

S

i

is a filter,

A

∩

B

∈

S

i

, hence

A

∩

B

∈

𝒰

. By the alternative

characterization

of a filter,

𝒰

is a filter.

Moreover,

𝒰

is an

ultrafilter

. Suppose that

A

∈

𝒰

and

B

∈

𝒰

are

disjoint

and

A

∪

B

=

X

. If either

A

∈

ℱ

or

B

∈

ℱ

, then either

A

∈

𝒰

or

B

∈

𝒰

because

ℱ

⊂

𝒰

. If

A

∈

Y

and

A

∈

S

A

, then

A

∈

𝒰

because

S

A

⊂

𝒰

. If

A

∈

Y

and

A

∉

S

A

, there must exist

x

∈

𝒰

such that

A

∩

x

is empty. Because

B

is the

complement

of

A

, this means that

x

⊂

B

and, hence

B

∈

𝒰

.

This

completes

the proof that

𝒰

is an ultrafilter — we have shown that

𝒰

meets the criteria given in the

alternative characterization of ultrafilters

.

Title

proof that every filter is contained in an ultrafilter

Canonical

name

ProofThatEveryFilterIsContainedInAnUltrafilter

Date of creation

2013-03-22 14:41:44

Last modified on

2013-03-22 14:41:44

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

17

Author

rspuzio (6075)

Entry type

Proof

Classification

msc 54A20

Generated on Sat Feb 10 11:14:27 2018 by

LaTeXML
theorem about continuous convergence

theorem about continuous convergence

Let

(

X

,

d

)

and

(

Y

,

ρ

)

be

metric spaces

, and let

f

n

:

X

⟶

Y

be a sequence of

continuous functions

. Then

f

n

converges continuously

to

f

if and only if

f

n

converges uniformly to

f

on every

compact subset

of

X

.

Title

theorem about continuous convergence

Canonical name

TheoremAboutContinuousConvergence

Date of creation

2013-03-22 14:05:00

Last modified on

2013-03-22 14:05:00

Owner

gumau (3545)

Last modified by

gumau (3545)

Numerical id

5

Author

gumau (3545)

Entry type

Theorem

Classification

msc 54A20

Generated on Sat Feb 10 11:14:28 2018 by

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ultrafilter

ultrafilter

Let

X

be a set.

Definitions

A

collection

𝒰

of subsets of

X

is an

ultrafilter

if

𝒰

is a filter, and whenever

A

⊆

X

then either

A

∈

𝒰

or

X

∖

A

∈

𝒰

.

Equivalently, an ultrafilter on

X

is a maximal (

http://planetmath.org/MaximalElement

) filter on

X

.

More generally, an ultrafilter of a

lattice

(

http://planetmath.org/LatticeFilter

)
is a maximal proper filter of the lattice.
This is indeed a

generalization

, as an ultrafilter on

X

can then be defined as an ultrafilter of the

power set

𝒫

⁢

(

X

)

.

Types of ultrafilter

For any

x

∈

X

the set

{

A

⊆

X

∣

x

∈

A

}

is an ultrafilter on

X

.
An ultrafilter formed in this way is called a

fixed ultrafilter

,
or a

principal ultrafilter

, or a

trivial ultrafilter

.
Any other ultrafilter on

X

is called a

free ultrafilter

,
or a

non-principal ultrafilter

.
An ultrafilter on a

finite set

is necessarily fixed.
On any

infinite set

there are free ultrafilters
(in great abundance (

http://planetmath.org/NumberOfUltrafilters

)),
but their

existence

depends on the

Axiom of Choice

,
and so none can be explicitly constructed.

An ultrafilter

𝒰

on

X

is called a

uniform ultrafilter

if every member of

𝒰

has the same

cardinality

.
(An ultrafilter on a

singleton

is uniform,
but this is a degenerate case and is often excluded.
All other uniform ultrafilters are free.)

Title

ultrafilter

Canonical

name

Ultrafilter

Date of creation

2013-03-22 12:13:59

Last modified on

2013-03-22 12:13:59

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

11

Author

yark (2760)

Entry type

Definition

Classification

msc 54A20

Related topic

Filter

Related topic

Ultranet

Related topic

EveryBoundedSequenceHasLimitAlongAnUltrafilter

Related topic

LatticeFilter

Defines

fixed ultrafilter

Defines

principal ultrafilter

Defines

trivial ultrafilter

Defines

free ultrafilter

Defines

non-principal ultrafilter

Defines

nonprincipal ultrafilter

Defines

uniform ultrafilter

Generated on Sat Feb 10 11:14:43 2018 by

LaTeXML
ultranet

ultranet

A net

(

x

a

)

a

∈

A

on a set

X

is said to be an

ultranet

or

universal net

if whenever

E

⊆

X

,

(

x

a

)

is either

eventually

in

E

or eventually in

X

∖

E

.

:

•

It can be shown that every net has a

universal

subnet.

•

When

X

is a

locally compact topological space

, a universal net in

X

is either convergent or it “goes to ” (it eventually leaves every

compact subset

).

Title

ultranet

Canonical name

Ultranet

Date of creation

2013-03-22 12:54:35

Last modified on

2013-03-22 12:54:35

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

9

Author

asteroid (17536)

Entry type

Definition

Classification

msc 54A20

Synonym

universal net

Related topic

Ultrafilter

Related topic

EveryNetHasAUniversalSubnet

Generated on Sat Feb 10 11:14:44 2018 by

LaTeXML
universal nets in compact spaces are convergent

universal nets in compact spaces are convergent

Theorem

-

A

universal net

(

x

α

)

α

∈

𝒜

in a

compact space

X

is convergent.

Proof :

Suppose by

contradiction

that

(

x

α

)

α

∈

𝒜

was not convergent. Then for every

x

∈

X

we would find

neighborhoods

U

x

such that

∀

α

∈

𝒜

∃

α

≤

α

0

x

α

0

∉

U

x

The

collection

of all this neighborhoods cover

X

, and as

X

is

compact

, a finite number

U

x

1

,

U

x

2

,

…

,

U

x

n

also cover

X

.

The net

(

x

α

)

α

∈

𝒜

is not eventually in

U

x

k

so it must be eventually in

X

-

U

x

k

(because it is a net). Therefore we can find

α

k

∈

𝒜

such that

∀

α

k

≤

α

x

α

∈

X

-

U

x

k

Because we have a finite number

α

1

,

α

2

⁢

…

,

α

n

∈

𝒜

we can find

γ

∈

𝒜

such that

α

k

≤

γ

for each

1

≤

k

≤

n

.

Then

x

γ

∈

X

-

U

x

k

for all

k

, i.e.

x

γ

∉

U

x

k

for all

k

. But

U

x

1

,

U

x

2

,

…

,

U

x

n

cover

X

and thus we have a contradiction.

□

Title

universal nets in compact spaces are convergent

Canonical name

UniversalNetsInCompactSpacesAreConvergent

Date of creation

2013-03-22 17:31:29

Last modified on

2013-03-22 17:31:29

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

4

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 54A20

Generated on Sat Feb 10 11:14:46 2018 by

LaTeXML
absolute retract

absolute retract

A

topological space

X

is an

absolute retract

if, for every

embedding

of

X

as a

closed subset

of a

normal space

Y

, the image of

X

is a

retract

of

Y

.

A topological space

X

is an if, for every embedding of

X

as a closed subset of a normal space

Y

, the image of

X

is a

neighborhood retract

of

Y

.

Title

absolute retract

Canonical name

AbsoluteRetract

Date of creation

2013-03-22 14:39:35

Last modified on

2013-03-22 14:39:35

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

5

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 54A99

Defines

absolute neighborhood retract

Generated on Sat Feb 10 11:14:48 2018 by

LaTeXML
adherent point

adherent point

Let

X

be a

topological space

and

A

⊂

X

be a subset. A point

x

∈

X

is an

adherent point

for

A

if every

open set

containing

x

contains at least one point of

A

. A point

x

is an adherent point for

A

if and only if

x

is in the

closure

of

A

.

Note that this

definition

is slightly more general than that of a

limit point

, in that for a limit point it is required that every open set containing

x

contains at least one point of

A

different from

x

.

References

1

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

Counterexamples in topology

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

Title

adherent point

Canonical

name

AdherentPoint

Date of creation

2013-03-22 14:38:18

Last modified on

2013-03-22 14:38:18

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 54A99

Generated on Sat Feb 10 11:14:49 2018 by

LaTeXML
basis (topology)

basis (topology)

Let

(

X

,

𝒯

)

be a

topological space

. A subset

ℬ

of

𝒯

is a

basis

for

𝒯

if every member of

𝒯

is a union of members of

ℬ

.

Equivalently,

ℬ

is a basis if and only if whenever

U

is open and

x

∈

U

then there is an

open set

V

∈

ℬ

such that

x

∈

V

⊆

U

.

The

topology

generated by a basis

ℬ

consists of exactly the unions of the elements of

ℬ

.

We also have the following easy

characterization

: (for a proof, see the attachment)

Proposition

.

A

collection

of subsets

B

of

X

is a basis for some topology on

X

if and only if each

x

∈

X

is in some

element

B

∈

B

and whenever

B

1

,

B

2

∈

B

and

x

∈

B

1

∩

B

2

then there is

B

3

∈

B

such that

x

∈

B

3

⊆

B

1

∩

B

2

.

0.0.1

Examples

1. A basis for the

usual topology

of the real line is given by the set of

open intervals

since every open set can be expressed as a union of open intervals. One may choose a smaller set as a basis. For

instance

, the set of all open intervals with

rational

endpoints

and the set of all intervals whose length is a power of

1

/

2

are also bases. However, the set of all open intervals of length

1

is not a basis although it is a

subbasis

(since any interval of length less than

1

can be expressed as an

intersection

of two intervals of length

1

).

2. More generally, the set of

open balls

forms a basis for the topology on a metric space.

3. The set of all subsets with one element forms a basis for the

discrete topology

on any set.

Title

basis (topology)

Canonical

name

Basistopology

Date of creation

2013-03-22 12:05:03

Last modified on

2013-03-22 12:05:03

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

16

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 54A99

Synonym

basis

Synonym

base

Synonym

topology generated by a basis

Related topic

Subbasis

Related topic

CompactMetricSpacesAreSecondCountable

Generated on Sat Feb 10 11:14:51 2018 by

LaTeXML
boundary of a closed set is nowhere dense

boundary of a closed set is nowhere dense

Let

A

be closed. In general, the

boundary

of a set is closed. So it suffices to show that

∂

⁡

A

has empty

interior

.

Let

U

⊂

∂

⁡

A

be open. Since

∂

⁡

A

⊂

A

¯

=

A

, this implies that

U

⊂

A

. Since

int

⁡

(

A

)

is the largest

open subset

of

A

, we must have

U

⊂

int

⁡

(

A

)

. Therefore

U

⊂

∂

⁡

A

∩

int

⁡

(

A

)

. But

∂

⁡

A

∩

int

⁡

(

A

)

=

(

A

¯

-

int

⁡

(

A

)

)

∩

int

⁡

(

A

)

=

∅

, so

U

=

∅

.

Title

boundary of a

closed set

is nowhere dense

Canonical name

BoundaryOfAClosedSetIsNowhereDense

Date of creation

2013-03-22 18:34:01

Last modified on

2013-03-22 18:34:01

Owner

neapol1s (9480)

Last modified by

neapol1s (9480)

Numerical id

4

Author

neapol1s (9480)

Entry type

Derivation

Classification

msc 54A99

Generated on Sat Feb 10 11:14:53 2018 by

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