aarjaneiro/planetmath huggingface dataset:
nested interval theorem

nested interval theorem

Proposition 1

.

If

[

a

1

,

b

1

]

⊇

[

a

2

,

b

2

]

⊇

[

a

3

,

b

3

]

⊇

…

is a

sequence

of nested

closed intervals

, then

⋂

n

=

1

∞

[

a

n

,

b

n

]

≠

∅

.

If also

lim

n

→

∞

⁡

(

b

n

-

a

n

)

=

0

,  then the infinite
intersection consists of a unique

real number

.

Proof.

There are two consequences to nesting of

intervals

:

[

a

m

,

b

m

]

⊆

[

a

n

,

b

n

]

for

n

≤

m

:

1.

first of all, we have the

inequality

a

n

≤

a

m

for

n

≤

m

, which means that the sequence

a

1

,

a

2

,

…

,

a

n

,

…

is nondecreasing;

2.

in

addition

, we also have two inequalities:

a

m

≤

b

n

and

a

n

≤

b

m

. In either case, we have that

a

i

≤

b

j

for all

i

,

j

. This means that the sequence

a

1

,

a

2

,

…

,

a

n

,

…

is

bounded from above

by all

b

i

, where

i

=

1

,

2

,

…

.

Therefore, the limit of the sequence

(

a

i

)

exists, and is just the supremum, say

a

(see proof here (

http://planetmath.org/NondecreasingSequenceWithUpperBound

)). Similarly the sequence

(

b

i

)

is nonincreasing and bounded from below by all

a

i

, where

i

=

1

,

2

,

…

, and hence has an infimum

b

.

Now, as the supremum of

(

a

i

)

,

a

≤

b

i

for all

i

. But because

b

is the infimum of

(

b

i

)

,

a

≤

b

. Therefore, the interval

[

a

,

b

]

is non-empty (containing at least one of

a

,

b

). Since

a

i

≤

a

≤

b

≤

b

i

, every interval

[

a

i

,

b

i

]

contains the interval

[

a

,

b

]

, so their intersection also contains

[

a

,

b

]

, hence is non-empty.

If

c

is a point outside of

[

a

,

b

]

, say

c

<

a

, then there is some

a

i

, such that

c

<

a

i

(by the definition of the supremum

a

), and hence

c

∉

[

a

i

,

b

i

]

. This shows that the intersection actually coincides with

[

a

,

b

]

.

Now, since

lim

n

→

∞

⁡

(

b

n

-

a

n

)

=

0

, we have that

b

-

a

=

lim

n

→

∞

⁡

b

n

-

lim

n

→

∞

⁡

a

n

=

lim

n

→

∞

⁡

(

b

n

-

a

n

)

=

0

. So

a

=

b

. This means that the intersection of the nested intervals contains a single point

a

.
∎

Remark

.  This result is called the

nested interval theorem

.
It is a restatement of the

finite intersection property

for the

compact set

[

a

1

,

b

1

]

.  The result may also be proven by

elementary methods

:
namely, any number lying in between the supremum of all the

a

n

and the infimum of all the

b

n

will be in all the nested intervals.

Title

nested interval theorem

Canonical name

NestedIntervalTheorem

Date of creation

2013-03-22 17:27:12

Last modified on

2013-03-22 17:27:12

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Theorem

Classification

msc 54C30

Classification

msc 26-00

Generated on Sat Feb 10 11:19:10 2018 by

LaTeXML
the groups of real numbers

the groups of real numbers

Proposition 1

.

The

additive group

of

real number

⟨

R

,

+

⟩

is

isomorphic

to the

multiplicative group

of positive real numbers

⟨

R

+

,

×

⟩

.

Proof.

Let

f

⁢

(

x

)

=

e

x

. This maps the group

⟨

ℝ

,

+

⟩

to the group

⟨

ℝ

+

,

×

⟩

. As

f

has an

inverse

f

-

1

⁢

(

x

)

=

ln

⁡

x

we observe

f

is

invertible

. Furthermore,

f

⁢

(

x

+

y

)

=

e

x

+

y

=

e

x

⁢

e

y

=

f

⁢

(

x

)

⁢

f

⁢

(

y

)

so

f

is a

homomorphism

. Thus

f

is an isomorphism.
∎

Corollary 2

.

The multiplicative group of non-zero real number

R

×

is isomorphic to

Z

2

⊕

⟨

R

,

+

⟩

.

Proof.

Use the map

f

:

ℤ

2

⊕

⟨

ℝ

,

+

⟩

→

ℝ

×

defined by

f

⁢

(

s

,

r

)

=

(

-

1

)

s

⁢

e

r

.

1

1

We write

(

-

1

)

s

to mean

(

-

1

)

s

′

for any integer

s

′

representative of the

equivalence class

of

s

in

ℤ

2

.

Then

f

⁢

(

(

s

1

,

r

1

)

+

(

s

2

,

r

2

)

)

=

f

⁢

(

s

1

+

s

2

,

r

1

+

r

2

)

=

(

-

1

)

s

1

+

s

2

⁢

e

r

1

+

r

2

=

(

-

1

)

s

1

⁢

e

r

1

⁢

(

-

1

)

s

2

⁢

e

r

2

=

f

⁢

(

s

1

,

r

1

)

⁢

f

⁢

(

s

2

,

r

2

)

so that

f

is a homomorphism. Furthermore,

f

-

1

⁢

(

r

)

=

(

sign

⁡

r

,

ln

⁡

|

r

|

)

is the inverse of

f

so that

f

is

bijective

and thus an isomorphism of groups.
∎

Title

the

groups of real numbers

Canonical name

TheGroupsOfRealNumbers

Date of creation

2013-03-22 16:08:50

Last modified on

2013-03-22 16:08:50

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

5

Author

Algeboy (12884)

Entry type

Result

Classification

msc 54C30

Classification

msc 26-00

Classification

msc 12D99

Related topic

ExponentialFunction

Related topic

GroupHomomorphism

Related topic

GroupsInField

Generated on Sat Feb 10 11:19:18 2018 by

LaTeXML
compactly supported continuous functions are dense in L^p

compactly supported continuous functions are dense in

L

p

Let

(

X

,

ℬ

,

μ

)

be a

measure space

, where

X

is a

locally compact Hausdorff space

,

ℬ

a

σ

-algebra (

http://planetmath.org/SigmaAlgebra

) that contains all

compact subsets

of

X

and

μ

a

measure

such that:

•

μ

⁢

(

K

)

<

∞

for all compact sets

K

⊂

X

.

•

μ

is

inner regular

, meaning

μ

⁢

(

A

)

=

sup

⁡

{

μ

⁢

(

K

)

:

K

⊂

A

,

K

⁢

is compact

}

•

μ

is outer regular, meaning

μ

⁢

(

A

)

=

inf

⁡

{

μ

⁢

(

U

)

:

A

⊂

U

,

U

∈

ℬ

⁢

and

⁢

U

⁢

is open

}

We denote by

C

c

⁢

(

X

)

the space of

continuous functions

X

→

ℂ

with compact support.

Theroem -

For every

1

≤

p

<

∞

,

C

c

⁢

(

X

)

is dense in

L

p

⁢

(

X

)

(

http://planetmath.org/LpSpace

).

:

It is clear that

C

c

⁢

(

X

)

is indeed

contained

in

L

p

⁢

(

X

)

, where we identify each

function

in

C

c

⁢

(

X

)

with its class in

L

p

⁢

(

X

)

.

We begin by proving that for each

A

∈

ℬ

with finite measure, the

characteristic function

χ

A

can be approximated, in the

L

p

norm, by functions in

C

c

⁢

(

X

)

. Let

ϵ

>

0

. By of

μ

, we know there exist an

open set

U

and a compact set

K

such that

K

⊂

A

⊂

U

and

μ

⁢

(

U

∖

K

)

=

μ

⁢

(

U

)

-

μ

⁢

(

K

)

<

ϵ

By the Urysohn’s lemma for locally compact Hausdorff spaces (

http://planetmath.org/ApplicationsOfUrysohnsLemmaToLocallyCompactHausdorffSpaces

), we know there is a function

f

∈

C

c

⁢

(

X

)

such that

0

≤

f

≤

1

,

f

|

K

=

1

and

supp

⁢

f

⊂

U

. Hence,

∫

X

|

χ

A

-

f

|

p

⁢

𝑑

μ

=

∫

U

∖

K

|

χ

A

-

f

|

p

⁢

𝑑

μ

<

ϵ

Thus,

χ

A

can be approximated in

L

p

by functions in

C

c

⁢

(

X

)

.

Now, it follows easily that any

simple function

∑

i

=

1

n

c

i

⁢

χ

A

i

, where each

A

i

has finite measure, can also be approximated by a compactly supported continuous function. Since this kind of simple functions are dense in

L

p

⁢

(

X

)

we see that

C

c

⁢

(

X

)

is also dense in

L

p

⁢

(

X

)

.

□

Title

compactly supported continuous functions are dense in

L

p

Canonical name

CompactlySupportedContinuousFunctionsAreDenseInLp

Date of creation

2013-03-22 18:38:53

Last modified on

2013-03-22 18:38:53

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

6

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 54C35

Classification

msc 46E30

Classification

msc 28C15

Synonym

C

c

⁢

(

X

)

is dense in

L

p

⁢

(

X

)

Generated on Sat Feb 10 11:19:22 2018 by

LaTeXML
C-embedding

C

-embedding

Let

X

be a

topological space

, and

C

⁢

(

X

)

the

ring of continuous functions

on

X

. A

subspace

A

⊆

X

is said to be

C

-embedded

(in

X

) if every

function

in

C

⁢

(

A

)

can be extended to a function in

C

⁢

(

X

)

. More precisely, for every real-valued

continuous function

f

:

A

→

ℝ

, there is a real-valued continuous function

g

:

X

→

ℝ

such that

g

⁢

(

x

)

=

f

⁢

(

x

)

for all

x

∈

A

.

If

A

⊆

X

is

C

-embedded,

f

↦

g

(defined above) is an

embedding

of

C

⁢

(

A

)

into

C

⁢

(

X

)

by

axiom of choice

, and hence the nomenclature.

Similarly, one may define

C

*

-embedding

on subspaces of a topological space. Recall that for a topological space

X

,

C

*

⁢

(

X

)

is the ring of

bounded

continuous functions on

X

. A subspace

A

⊆

X

is said to be

C

*

-embedded

(in

X

) if every

f

∈

C

*

⁢

(

A

)

can be extended to some

g

∈

C

*

⁢

(

X

)

.

Remarks

. Let

A

be a subspace of

X

.

1.

If

A

is

C

-embedded in

X

, and

A

⊆

Y

⊆

X

, then

A

is

C

-embedded in

Y

. This is also true for

C

*

-embeddedness.

2.

If

A

is

C

-embedded, then

A

is

C

*

-embedded: for if

f

is a bounded continuous function on

A

, say

-

n

≤

f

≤

n

, and

g

is its

continuous

extension

on

X

, then

-

n

∨

(

g

∧

n

)

is a bounded continuous extension of

f

on

X

.

3.

The

converse

, however, is not true in general. A necessary and

sufficient

condition that a

C

*

-embedded set

A

is

C

-embedded is:

if a

zero set

is

disjoint

from

A

, the it is

completely separated

from

A

.

Since any pair of disjoint zero sets are completely separated, we have that if

A

is a

C

*

-embedded zero set, then

A

is

C

-embedded.

References

1

L. Gillman, M. Jerison:

Rings of Continuous Functions

, Van Nostrand, (1960).

Title

C

-embedding

Canonical name

Cembedding

Date of creation

2013-03-22 16:57:37

Last modified on

2013-03-22 16:57:37

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 54C45

Synonym

C-embedded

Synonym

C embedded

Synonym

C*-embedded

Synonym

C* embedded

Defines

C

-embedded

Defines

C

*

-embedded

Generated on Sat Feb 10 11:19:44 2018 by

LaTeXML
Urysohn extension theorem

Urysohn extension theorem

Let

X

be a

topological space

, and

C

⁢

(

X

)

and

C

*

⁢

(

X

)

the

rings of continuous functions

and

bounded

continuous functions

respectively.

Urysohn Extension Theorem

.

A subset

A

⊆

X

is

C

*

-embedded

(

http://planetmath.org/CEmbedding

)

if and only if any two

completely separated

sets in

A

are completely separated in

X

as well.

Remarks

.

•

Suppose that

X

is a

metric space

and

A

is closed in

X

. If

S

,

T

are completely separated sets in

A

, then they are

contained

in

disjoint

zero sets

S

′

and

T

′

in

A

. Since

S

′

and

T

′

are closed in

A

, and

A

is closed in

X

,

S

′

and

T

′

are closed in

X

. Since

X

is a metric space,

S

′

and

T

′

are zero sets in

X

. Since

S

⊆

S

′

and

T

⊆

T

′

and

S

′

∩

T

′

are disjoint,

S

and

T

are completely separated in

X

as well. By Urysohn Extension Theorem, any bounded continuous function defined on

A

can be extended to a continuous function on

X

, which is the statement of the metric space version of the

Tietze extension theorem

.

•

However, the above

argument

does not generalize to

normal spaces

, so can not be used to prove the generalized version of the Tietze extension theorem. Urysohn’s lemma is required to prove this more general result.

Title

Urysohn extension theorem

Canonical name

UrysohnExtensionTheorem

Date of creation

2013-03-22 17:01:43

Last modified on

2013-03-22 17:01:43

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 54C45

Generated on Sat Feb 10 11:19:52 2018 by

LaTeXML
differential entropy

differential entropy

Let

(

X

,

𝔅

,

μ

)

be a

probability space

, and let

f

∈

L

p

⁢

(

X

,

𝔅

,

μ

)

,

||

f

||

p

=

1

be a

function

. The

differential entropy

h

⁢

(

f

)

is defined as

h

⁢

(

f

)

≡

-

∫

X

|

f

|

p

⁢

log

⁡

|

f

|

p

⁢

d

⁢

μ

(1)

Differential entropy is the continuous version of the

Shannon entropy

,

H

⁢

[

𝐩

]

=

-

∑

i

p

i

⁢

log

⁡

p

i

. Consider first

u

a

, the uniform 1-dimensional

distribution

on

(

0

,

a

)

. The differential entropy is

h

⁢

(

u

a

)

=

-

∫

0

a

1

a

⁢

log

⁡

1

a

⁢

d

⁢

μ

=

log

⁡

a

.

(2)

Next consider probability distributions such as the function

g

=

1

2

⁢

π

⁢

σ

⁢

e

-

(

t

-

μ

)

2

2

⁢

σ

2

,

(3)

the 1-dimensional

Gaussian

. This pdf has differential entropy

h

⁢

(

g

)

=

-

∫

ℝ

g

⁢

log

⁡

g

⁢

d

⁢

t

=

1

2

⁢

log

⁡

2

⁢

π

⁢

e

⁢

σ

2

.

(4)

For a general

n

-dimensional Gaussian (

http://planetmath.org/JointNormalDistribution

)

𝒩

n

⁢

(

μ

,

𝐊

)

with

mean vector

μ

and

covariance matrix

𝐊

,

K

i

⁢

j

=

cov

⁢

(

x

i

,

x

j

)

, we have

h

(

𝒩

n

(

μ

,

𝐊

)

)

=

1

2

log

(

2

π

e

)

n

|

𝐊

|

(5)

where

|

𝐊

|

=

det

⁡

𝐊

.

Title

differential entropy

Canonical name

DifferentialEntropy

Date of creation

2013-03-22 12:18:48

Last modified on

2013-03-22 12:18:48

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

16

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 54C70

Related topic

ShannonsTheoremEntropy

Related topic

ConditionalEntropy

Generated on Sat Feb 10 11:20:05 2018 by

LaTeXML
Borsuk-Ulam theorem

Borsuk-Ulam theorem

Call a

continuous map

f

:

S

m

→

S

n

antipode preserving if

f

⁢

(

-

x

)

=

-

f

⁢

(

x

)

for all

x

∈

S

m

.

Theorem

: There exists no continuous map

f

:

S

n

→

S

n

-

1

which is antipode preserving for

n

>

0

.

Some interesting consequences of this theorem have real-world applications. For example, this theorem implies that at any time there exists antipodal points on the surface of the earth which have exactly the same barometric pressure and temperature.

It is also interesting to note a corollary to this theorem which states that no subset of

ℝ

n

is

homeomorphic

to

S

n

.

Title

Borsuk-Ulam theorem

Canonical name

BorsukUlamTheorem

Date of creation

2013-03-22 12:00:18

Last modified on

2013-03-22 12:00:18

Owner

RevBobo (4)

Last modified by

RevBobo (4)

Numerical id

7

Author

RevBobo (4)

Entry type

Theorem

Classification

msc 54C99

Related topic

HamSandwichTheorem

Generated on Sat Feb 10 11:20:07 2018 by

LaTeXML
ham sandwich theorem

ham sandwich theorem

Let

A

1

,

…

,

A

m

be measurable

bounded

subsets of

ℝ

m

. Then there exists an

(

m

-

1

)

-dimensional

hyperplane

which each

A

i

into two subsets of equal measure.

This theorem has such a colorful because in the case

m

=

3

it can be viewed as

cutting

a ham sandwich in half. For example,

A

1

and

A

3

could be two pieces of bread and

A

2

a piece of ham. According to this theorem it is possible to make one to simultaneously all three objects exactly in half.

Title

ham sandwich theorem

Canonical name

HamSandwichTheorem

Date of creation

2013-03-22 13:59:43

Last modified on

2013-03-22 13:59:43

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 54C99

Related topic

BorsukUlamTheorem

Generated on Sat Feb 10 11:20:08 2018 by

LaTeXML
a connected and locally path connected space is path connected

a connected and locally path connected space is path connected

Theorem. A

connected

,

locally path connected

topological space

is

path connected

.

Proof. Let

X

be the space and fix

p

∈

X

. Let

C

be the set of all points in

X

that can be joined to

p

by a path.

C

is nonempty so it is enough to show that

C

is both

closed and open

.

To show first that

C

is open: Let

c

be in

C

and choose an

open path

connected

neighborhood

U

of

c

. If

u

∈

U

we can find a path joining

u

to

c

and then join that path to a path from

p

to

c

. Hence

u

is in

C

.

To show that

C

is closed: Let

c

be in

C

¯

and choose an open path connected neighborhood

U

of

c

. Then

C

∩

U

≠

∅

. Choose

q

∈

C

∩

U

. Then

c

can be joined to

q

by a path and

q

can be joined to

p

by a path, so by addition of paths,

p

can be joined to

c

by a path, that is,

c

∈

C

.

Title

a

connected and locally path connected space is path connected

Canonical name

AConnectedAndLocallyPathConnectedSpaceIsPathConnected

Date of creation

2013-03-22 16:50:43

Last modified on

2013-03-22 16:50:43

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

6

Author

Mathprof (13753)

Entry type

Theorem

Classification

msc 54D05

Generated on Sat Feb 10 11:20:16 2018 by

LaTeXML
a connected normal space with more than one point is uncountable

a connected normal space with more than one point is uncountable

The proof of the following result is an application of the

generalized intermediate value theorem

(along with Urysohn’s lemma):

Proposition

.

A

connected normal space with more than one point is uncountable

.

Proof.

Let

X

be a

http://planetmath.org/node/941

connected

http://planetmath.org/node/1532

normal space

with at least two distinct points

x

1

and

x

2

. As the sets

{

x

1

}

and

{

x

2

}

are

http://planetmath.org/node/2739

closed and

disjoint

, Urysohn’s lemma furnishes a

continuous function

f

:

X

→

[

0

,

1

]

such that

f

⁢

(

x

1

)

=

0

and

f

⁢

(

x

2

)

=

1

. Because

X

is connected, the generalized intermediate value theorem implies that

f

is

surjective

. Thus

f

may be suitably to give a

bijection

between a subset of

X

and the

uncountable set

[

0

,

1

]

, from which it follows that

X

is uncountable.
∎

Title

a connected normal space with more than one point is uncountable

Canonical name

AConnectedNormalSpaceWithMoreThanOnePointIsUncountable

Date of creation

2013-03-22 17:17:46

Last modified on

2013-03-22 17:17:46

Owner

azdbacks4234 (14155)

Last modified by

azdbacks4234 (14155)

Numerical id

10

Author

azdbacks4234 (14155)

Entry type

Theorem

Classification

msc 54D05

Related topic

UrysohnsLemma

Related topic

NormalTopologicalSpace

Related topic

Uncountable

Related topic

Bijection

Related topic

ConnectedSpace

Generated on Sat Feb 10 11:20:18 2018 by

LaTeXML
balls in ultrametric spaces are clopen subsets

balls in ultrametric spaces are clopen subsets

In an

ultrametric space

, both open and

closed balls

are

clopen subsets

.

It is indeed straightforward (exercise!) to show that the set of all open balls of radius

r

,
centered in any of the points of a closed ball of radius

r

, forms a

partition

of the latter.

Thus, in particular, any point of a closed ball is an

interior point

,
and the same holds for the

complement

of an open ball.

Title

balls in ultrametric spaces are clopen subsets

Canonical name

BallsInUltrametricSpacesAreClopenSubsets

Date of creation

2013-03-22 18:20:15

Last modified on

2013-03-22 18:20:15

Owner

MFH (21412)

Last modified by

MFH (21412)

Numerical id

5

Author

MFH (21412)

Entry type

Example

Classification

msc 54D05

Generated on Sat Feb 10 11:20:19 2018 by

LaTeXML
clopen subset

clopen subset

A subset of a

topological space

X

is called

clopen

if it is both
open and closed.

Theorem 1

.

The

clopen subsets

form a

Boolean algebra

under the

operation

of
union,

intersection

and

complement

. In other words:

•

X

and

∅

are clopen,

•

the complement of a clopen set is clopen,

•

finite unions and intersections of clopen sets are clopen.

Proof.

The first follows by the

definition

of a

topology

, the second by
noting that complements of

open sets

are closed, and vice versa,
and the third by noting that this

property

holds for both open
and

closed sets

.
∎

One application of clopen sets is that they can be used to
describe connectness. In particular, a topological space is

connected

if and only if its only clopen subsets are itself and
the

empty set

.

If a space has finitely many

connected components

then each
connected component is clopen. This may not be the case if there
are infinitely many components, as the case of the

rational numbers

demonstrates.

Title

clopen subset

Canonical

name

ClopenSubset

Date of creation

2013-03-22 13:25:29

Last modified on

2013-03-22 13:25:29

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

14

Author

mathcam (2727)

Entry type

Definition

Classification

msc 54D05

Synonym

clopen set

Synonym

clopen

Synonym

closed and open

Related topic

IdentityTheorem

Generated on Sat Feb 10 11:20:21 2018 by

LaTeXML
connected component

connected component

Two points

x

,

y

in a

topological space

X

are said to be in the same

connected component

if there exists a

subspace

of

X

containing

x

and

y

which is

connected

. This

relation

is an

equivalence relation

, and the

equivalence classes

of

X

under this relation are called the

connected components

of

X

.

Title

connected component

Canonical name

ConnectedComponent

Date of creation

2013-03-22 12:00:21

Last modified on

2013-03-22 12:00:21

Owner

djao (24)

Last modified by

djao (24)

Numerical id

6

Author

djao (24)

Entry type

Definition

Classification

msc 54D05

Generated on Sat Feb 10 11:20:22 2018 by

LaTeXML
connected im kleinen

connected im kleinen

A

topological space

X

is

connected im kleinen

at a point

x

if every

open set

U

containing

x

contains an open set

V

containing

x

such that if

y

is a point of

V

, then there is a

connected

subset of

U

containing

{

x

,

y

}

.

Another way to say this is that

X

is connected im kleinen at a point

x

if

x

has a

neighborhood base

of connected sets (not necessarily open).

A locally connected space is connected im kleinen at each point.

A space can be connected im kleinen at a point but not

locally connected

at the point.

If a topological space is connected im kleinen at each point, then it is locally
connected.

References

1

S. Willard,

General Topology

,
Addison-Wesley, Publishing Company, 1970.

2

J.G. Hocking, G.S. Young,

Topology

, Dover Pubs,
1988, republication of 1961 Addison-Wesley edition.

Title

connected im kleinen

Canonical name

ConnectedImKleinen

Date of creation

2013-03-22 15:59:00

Last modified on

2013-03-22 15:59:00

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

8

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 54D05

Generated on Sat Feb 10 11:20:24 2018 by

LaTeXML
connected set in a topological space

connected set in a topological space

Let

Y

be a

topological space

and

X

⊆

Y

be given the

subspace topology

. We say that

X

is

connected

iff we cannot find nonempty

open sets

U

,

V

⊆

X

such that

U

∩

V

=

∅

and

U

∪

V

=

X

.

Title

connected set in a topological space

Canonical name

ConnectedSetInATopologicalSpace

Date of creation

2013-03-22 13:59:51

Last modified on

2013-03-22 13:59:51

Owner

ack (3732)

Last modified by

ack (3732)

Numerical id

6

Author

ack (3732)

Entry type

Definition

Classification

msc 54D05

Generated on Sat Feb 10 11:20:26 2018 by

LaTeXML