aarjaneiro/planetmath huggingface dataset:
constant function

constant function

Definition

Suppose

X

and

Y

are sets and

f

:

X

→

Y

is a

function

.
Then

f

is a

constant function

if

f

⁢

(

a

)

=

f

⁢

(

b

)

for all

a

,

b

in

X

.

0.0.1

Properties

1.

The

composition

of a constant function
with any function (for which composition is defined) is a constant function.

2.

A

constant map

between

topological spaces

is

continuous

.

Title

constant function

Canonical

name

ConstantFunction

Date of creation

2013-03-22 13:57:35

Last modified on

2013-03-22 13:57:35

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

5

Author

mathcam (2727)

Entry type

Definition

Classification

msc 03E20

Synonym

constant function

Synonym

constant map

Synonym

constant mapping

Related topic

Function

Related topic

HomotopyOfMaps

Related topic

ZeroMap

Generated on Thu Feb 8 20:12:23 2018 by

LaTeXML
constant functions and continuity

constant functions and continuity

It is easy to see that every

constant function

between

topological
spaces

is

continuous

. A

converse

result is as follows.

Theorem

.

Suppose

X

is

path connected

and

D

is a

countable

discrete topological space

. If

f

:

X

→

D

is continuous,
then

f

is a constant function.

Proof.

By this result (

http://planetmath.org/FiniteAndCountableDiscreteSpaces

)
we can assume that

D

is either

{

1

,

…

,

n

}

,

n

≥

2

or

ℤ

, and these are equipped with the

subspace topology

of

ℝ

.
Suppose

f

⁢

(

X

)

has at least two distinct

elements

, say

α

,

β

∈

ℤ

so that

f

⁢

(

x

)

=

α

,

f

⁢

(

y

)

=

β

for some

x

,

y

∈

X

. Since

X

is path connected there is a continuous path

γ

:

[

0

,

1

]

→

X

such that

γ

⁢

(

0

)

=

x

and

γ

⁢

(

1

)

=

y

.
Then

f

∘

γ

:

[

0

,

1

]

→

D

is continuous.
Since

D

has the subspace topology of

ℝ

,
this result (

http://planetmath.org/ContinuityIsPreservedWhenCodomainIsExtended

)
implies that
also

f

∘

γ

:

[

0

,

1

]

→

ℝ

is continuous.
Since

f

∘

γ

achieves two different values, it achieves uncountably many values,
by the

intermediate value theorem

.
This is a

contradiction

since

f

∘

γ

⁢

(

[

0

,

1

]

)

is countable.
∎

Title

constant functions and continuity

Canonical name

ConstantFunctionsAndContinuity

Date of creation

2013-03-22 15:17:31

Last modified on

2013-03-22 15:17:31

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

12

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 03E20

Generated on Thu Feb 8 20:12:25 2018 by

LaTeXML
criterion for a set to be transitive

criterion for a set to be transitive

Theorem

.

A set

X

is

transitive

if and only if its

power set

P

⁢

(

X

)

is transitive.

Proof.

First assume

X

is transitive. Let

A

∈

B

∈

𝒫

⁢

(

X

)

. Since

B

∈

𝒫

⁢

(

X

)

,

B

⊆

X

. Thus,

A

∈

X

. Since

X

is transitive,

A

⊆

X

. Hence,

A

∈

𝒫

⁢

(

X

)

. It follows that

𝒫

⁢

(

X

)

is transitive.

Conversely

, assume

𝒫

⁢

(

X

)

is transitive. Let

a

∈

X

. Then

{

a

}

∈

𝒫

⁢

(

X

)

. Since

𝒫

⁢

(

X

)

is transitive,

{

a

}

⊆

𝒫

⁢

(

X

)

. Thus,

a

∈

𝒫

⁢

(

X

)

. Hence,

a

⊆

X

. It follows that

X

is transitive.
∎

Title

criterion for a set to be transitive

Canonical name

CriterionForASetToBeTransitive

Date of creation

2013-03-22 16:18:23

Last modified on

2013-03-22 16:18:23

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

6

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 03E20

Related topic

CumulativeHierarchy

Generated on Thu Feb 8 20:12:28 2018 by

LaTeXML
dependence relation

dependence relation

Let

X

be a set. A (binary)

relation

≺

between an

element

a

of

X

and a subset

S

of

X

is called a

dependence relation

, written

a

≺

S

, when the following conditions are satisfied:

1.

if

a

∈

S

, then

a

≺

S

;

2.

if

a

≺

S

, then there is a finite subset

S

0

of

S

, such that

a

≺

S

0

;

3.

if

T

is a subset of

X

such that

b

∈

S

implies

b

≺

T

, then

a

≺

S

implies

a

≺

T

;

4.

if

a

≺

S

but

a

⊀

S

-

{

b

}

for some

b

∈

S

, then

b

≺

(

S

-

{

b

}

)

∪

{

a

}

.

Given a

dependence relation

≺

on

X

, a subset

S

of

X

is said to be

independent

if

a

⊀

S

-

{

a

}

for all

a

∈

S

. If

S

⊆

T

, then

S

is said to

span

T

if

t

≺

S

for every

t

∈

T

.

S

is said to be a

basis

of

X

if

S

is

independent

and

S

spans

X

.

Remark.

If

X

is a non-empty set with a

dependence relation

≺

, then

X

always has a

basis

with respect to

≺

. Furthermore, any two of

X

have the same

cardinality

.

Examples:

•

Let

V

be a

vector space

over a field

F

. The relation

≺

, defined by

υ

≺

S

if

υ

is in the

subspace

S

, is a dependence relatoin. This is

equivalent

to the definition of

linear dependence

(

http://planetmath.org/LinearIndependence

)

.

•

Let

K

be a field extension of

F

. Define

≺

by

α

≺

S

if

α

is algebraic over

F

⁢

(

S

)

. Then

≺

is a dependence relation. This is equivalent to the definition of

algebraic dependence

.

Title

dependence relation

Canonical name

DependenceRelation

Date of creation

2013-03-22 14:19:25

Last modified on

2013-03-22 14:19:25

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03E20

Classification

msc 05B35

Related topic

LinearIndependence

Related topic

AlgebraicallyDependent

Related topic

Matroid

Related topic

AxiomatizationOfDependence

Generated on Thu Feb 8 20:12:30 2018 by

LaTeXML
direct image

direct image

Let

f

:

A

⟶

B

be a

function

, and let

U

⊂

A

be a subset. The

direct image

of

U

is the set

f

⁢

(

U

)

⊂

B

consisting of all elements of

B

which equal

f

⁢

(

u

)

for some

u

∈

U

.

Direct images

satisfy

the following

properties

:

1.

Unions: For any

collection

{

U

i

}

i

∈

I

of subsets of

A

,

f

⁢

(

⋃

i

∈

I

U

i

)

=

⋃

i

∈

I

f

⁢

(

U

i

)

.

2.

Intersections

: For any collection

{

U

i

}

i

∈

I

of subsets of

A

,

f

⁢

(

⋂

i

∈

I

U

i

)

⊂

⋂

i

∈

I

f

⁢

(

U

i

)

.

3.

Set difference

: For any

U

,

V

⊂

A

,

f

⁢

(

V

∖

U

)

⊃

f

⁢

(

V

)

∖

f

⁢

(

U

)

.

In particular, the

complement

of

U

satisfies

f

⁢

(

U

∁

)

⊃

f

⁢

(

A

)

∖

f

⁢

(

U

)

.

4.

Subsets: If

U

⊂

V

⊂

A

, then

f

⁢

(

U

)

⊂

f

⁢

(

V

)

⊂

B

.

5.

Inverse image

of a direct image: For any

U

⊂

A

,

f

-

1

⁢

(

f

⁢

(

U

)

)

⊃

U

with equality if

f

is

injective

.

6.

Direct image of an inverse image: For any

V

⊂

B

,

f

⁢

(

f

-

1

⁢

(

V

)

)

⊂

V

with equality if

f

is

surjective

.

Title

direct image

Canonical

name

DirectImage

Date of creation

2013-03-22 11:52:01

Last modified on

2013-03-22 11:52:01

Owner

djao (24)

Last modified by

djao (24)

Numerical id

10

Author

djao (24)

Entry type

Definition

Classification

msc 03E20

Classification

msc 81-00

Classification

msc 18-00

Classification

msc 17B37

Classification

msc 18D10

Classification

msc 18D35

Classification

msc 16W30

Synonym

image

Related topic

InverseImage

Related topic

Mapping

Generated on Thu Feb 8 20:12:32 2018 by

LaTeXML
domain

domain

Let

R

be a

binary relation

. Then the set of all

x

such that

x

⁢

R

⁢

y

is called the

domain

of

R

. That is, the domain of

R

is the set of all first coordinates of the

ordered pairs

in

R

.

Title

domain

Canonical name

Domain1

Date of creation

2013-03-22 12:00:48

Last modified on

2013-03-22 12:00:48

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

9

Author

akrowne (2)

Entry type

Definition

Classification

msc 03E20

Generated on Thu Feb 8 20:12:34 2018 by

LaTeXML
Dynkin system

Dynkin system

Let

Ω

be a set, and

𝒫

⁢

(

Ω

)

be the

power set

of

Ω

. A

Dynkin system

on

Ω

is a set

𝒟

⊂

𝒫

⁢

(

Ω

)

such that

1.

Ω

∈

𝒟

2.

A

,

B

∈

𝒟

⁢

and

⁢

A

⊂

B

⇒

B

∖

A

∈

𝒟

3.

A

n

∈

𝒟

,

A

n

⊂

A

n

+

1

,

n

≥

1

⇒

⋃

k

=

1

∞

A

k

∈

𝒟

.

Let

F

⊂

𝒫

⁢

(

Ω

)

, and consider

Γ

=

{

X

:

X

⊂

𝒫

⁢

(

Ω

)

⁢

is a Dynkin system and

⁢

F

⊂

X

}

.

(1)

We define the

intersection

of all the Dynkin systems containing

F

as

𝒟

⁢

(

F

)

:=

⋂

X

∈

Γ

X

(2)

One can easily verify that

𝒟

⁢

(

F

)

is itself a Dynkin system and that it contains

F

. We call

𝒟

⁢

(

F

)

the

Dynkin system generated by

F

. It is the “smallest” Dynkin system containing

F

.

A Dynkin system which is also

π

-system (

http://planetmath.org/PiSystem

) is a

σ

-

algebra

(

http://planetmath.org/SigmaAlgebra

).

Title

Dynkin system

Canonical

name

DynkinSystem

Date of creation

2013-03-22 12:21:19

Last modified on

2013-03-22 12:21:19

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

9

Author

mathwizard (128)

Entry type

Definition

Classification

msc 03E20

Classification

msc 28A60

Related topic

DynkinsLemma

Generated on Thu Feb 8 20:12:36 2018 by

LaTeXML
equivalence class

equivalence class

Let

S

be a set with an

equivalence relation

∼

. An

equivalence class

of

S

under

∼

is a subset

T

⊂

S

such that

•

If

x

∈

T

and

y

∈

S

, then

x

∼

y

if and only if

y

∈

T

•

If

S

is nonempty, then

T

is nonempty

For

x

∈

S

, the equivalence class containing

x

is often denoted by

[

x

]

, so that

[

x

]

:=

{

y

∈

S

∣

x

∼

y

}

.

The

set of all equivalence classes

of

S

under

∼

is defined to be the set of all subsets of

S

which are equivalence classes of

S

under

∼

, and is denoted by

S

/

∼

. The map

x

↦

[

x

]

is sometimes referred to as the .

For any equivalence relation

∼

, the set of all equivalence classes of

S

under

∼

is a

partition

of

S

, and this correspondence is a

bijection

between the set of equivalence relations on

S

and the set of partitions of

S

(consisting of

nonempty sets

).

Title

equivalence class

Canonical name

EquivalenceClass

Date of creation

2013-03-22 11:52:30

Last modified on

2013-03-22 11:52:30

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

10

Author

mathcam (2727)

Entry type

Definition

Classification

msc 03E20

Classification

msc 93D05

Classification

msc 03B52

Classification

msc 93C42

Related topic

EquivalenceRelation

Related topic

Equivalent

Related topic

Partition

Generated on Thu Feb 8 20:12:38 2018 by

LaTeXML
example of antisymmetric

example of antisymmetric

The axioms of a

partial ordering

demonstrate that every partial ordering is antisymmetric. That is: the

relation

≤

on a set

S

forces

a

≤

b

and

b

≤

a

implies

a

=

b

for every

a

,

b

∈

S

.

For a concrete example consider the

natural numbers

ℕ

=

{

0

,

1

,

2

,

…

}

(as defined by the Peano postulates (

http://planetmath.org/PeanoArithmetic

)). Take the relation set to be:

R

=

{

(

a

,

a

+

n

)

:

a

,

n

∈

ℕ

}

⊂

ℕ

×

ℕ

.

Then we denote

a

≤

b

if

(

a

,

b

)

∈

R

. That is,

5

≤

7

because

(

5

,

7

)

=

(

5

,

5

+

2

)

and both

5

,

2

∈

ℕ

.

We can prove this relation is antisymmetric as follows: Suppose

a

≤

b

and

b

≤

a

for some

a

,

b

∈

ℕ

. Then there exist

n

,

m

∈

ℕ

such that

a

+

n

=

b

and

b

+

m

=

a

. Therefore

b

=

a

+

n

=

b

+

m

+

n

so by the cancellation property of the natural numbers,

0

=

m

+

n

. But by the first piano

postulate

, 0 has no

predecessor

, meaning

0

≠

m

+

n

unless

m

=

n

=

0

.

Title

example of antisymmetric

Canonical name

ExampleOfAntisymmetric

Date of creation

2013-03-22 16:00:36

Last modified on

2013-03-22 16:00:36

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

8

Author

Algeboy (12884)

Entry type

Example

Classification

msc 03E20

Generated on Thu Feb 8 20:12:40 2018 by

LaTeXML
examples of ring of sets

examples of ring of sets

Every

field of sets

is a ring of sets. Below are some

examples of rings of sets

that are not fields of sets.

1.

Let

A

be a non-empty set containing an

element

a

. Let

ℛ

be the family of subsets of

A

containing

a

. Then

ℛ

is a ring of sets, but not a field of sets, since

{

a

}

∈

ℛ

, but

A

-

{

a

}

∉

ℛ

.

2.

The

collection

of all

open sets

of a

topological space

is a ring of sets, which is in general not a field of sets, unless every open set is also closed. Likewise, the collection of all

closed sets

of a topological space is also a ring of sets.

3.

A simple example of a ring of sets is the subset

{

{

a

}

,

{

a

,

b

}

}

of

2

{

a

,

b

}

. That this is a ring of sets follows from the
observations that

{

a

}

∩

{

a

,

b

}

=

{

a

}

and

{

a

}

∪

{

a

,

b

}

=

{

a

,

b

}

. Note that it is not a field of sets because the

complement

of

{

a

}

, which is

{

b

}

, does not belong to the ring.

4.

Another example involves an

infinite set

. Let

A

be an infinite set. Let

ℛ

be the collection of finite subsets of

A

. Since the union and the

intersection

of two

finite set

are finite sets,

ℛ

is a ring of sets. However, it is not a field of sets, because the complement of a finite subset of

A

is

infinite

, and thus not a member of

ℛ

.

Title

examples of ring of sets

Canonical name

ExamplesOfRingOfSets

Date of creation

2013-03-22 15:47:52

Last modified on

2013-03-22 15:47:52

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

8

Author

rspuzio (6075)

Entry type

Example

Classification

msc 03E20

Classification

msc 28A05

Generated on Thu Feb 8 20:12:42 2018 by

LaTeXML
extension of a function

extension of a function

Let

f

:

X

→

Y

be a

function

and

A

and

B

be sets such that

X

⊆

A

and

Y

⊆

B

. An

extension

of

f

to

A

is a function

g

:

A

→

B

such that

f

⁢

(

x

)

=

g

⁢

(

x

)

for all

x

∈

X

. Alternatively,

g

is an extension of

f

to

A

if

f

is the

restriction

of

g

to

X

.

Typically, functions are not arbitrarily extended. Rather, it is usually insisted upon that extensions have certain

properties

. Examples include

analytic continuations

and

meromorphic extensions

.

Title

extension of a function

Canonical

name

ExtensionOfAFunction

Date of creation

2013-03-22 17:51:00

Last modified on

2013-03-22 17:51:00

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

6

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 03E20

Related topic

RestrictionOfAFunction

Defines

extension

Generated on Thu Feb 8 20:12:45 2018 by

LaTeXML
fibre

fibre

Given a

function

f

:

X

⟶

Y

, a

fibre

is an

inverse image

of an element of

Y

. That is given

y

∈

Y

,

f

-

1

⁢

(

{

y

}

)

=

{

x

∈

X

∣

f

⁢

(

x

)

=

y

}

is a fibre.

Example:

Define

f

:

ℝ

2

⟶

ℝ

by

f

⁢

(

x

,

y

)

=

x

2

+

y

2

. Then the fibres of

f

consist of

concentric circles

about the origin, the origin itself, and

empty sets

depending on whether we look at the inverse image of a positive number, zero, or a

negative number

respectively.

Example:

Suppose

M

is a

manifold

, and

π

:

T

⁢

M

→

M

is the
canonical projection from the

tangent bundle

T

⁢

M

to

M

. Then
fibres of

π

are the

tangent spaces

T

x

⁢

(

M

)

for

x

∈

M

.

Title

fibre

Canonical

name

Fibre

Date of creation

2013-03-22 12:55:23

Last modified on

2013-03-22 12:55:23

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

8

Author

mathcam (2727)

Entry type

Definition

Classification

msc 03E20

Synonym

fiber

Related topic

LevelSet

Generated on Thu Feb 8 20:12:47 2018 by

LaTeXML
filtration

filtration

A

filtration

is a

sequence

of sets

A

1

,

A

2

,

…

,

A

n

with

A

1

⊂

A

2

⊂

⋯

⊂

A

n

.

If one considers the sets

A

1

,

…

,

A

n

as elements of a larger set which are partially ordered by

inclusion

, then a filtration is simply a

finite chain

with respect to this

partial ordering

. It should be noted that in some

contexts

the word “filtration” may also be employed to describe an infinite chain.

Title

filtration

Canonical name

Filtration

Date of creation

2013-03-22 12:08:38

Last modified on

2013-03-22 12:08:38

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03E20

Related topic

FiltrationOfSigmaAlgebras

Generated on Thu Feb 8 20:12:48 2018 by

LaTeXML
finite character

finite character

A family

ℱ

of sets is of

finite character

if

1.

For each

A

∈

ℱ

, every finite subset of

A

belongs to

ℱ

;

2.

If every finite subset of a given set

A

belongs to

ℱ

, then

A

belongs to

ℱ

.

Title

finite character

Canonical name

FiniteCharacter

Date of creation

2013-03-22 13:13:31

Last modified on

2013-03-22 13:13:31

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

7

Author

Koro (127)

Entry type

Definition

Classification

msc 03E20

Generated on Thu Feb 8 20:12:50 2018 by

LaTeXML
fix (transformation action)

fix (transformation action)

Let

A

be a set, and

T

:

A

→

A

a

transformation

of that
set. We say that

x

∈

A

is

fixed

by

T

, or that

T

fixes

x

, whenever

T

⁢

(

x

)

=

x

.

The subset of fixed elements is called

the

fixed set

of

T

, and is frequently denoted as

A

T

.

We say that a subset

B

⊂

A

is
fixed by

T

whenever all elements of

B

are fixed by

T

,
i.e.

B

⊂

A

T

.

If this is so,

T

restricts to the

identity
transformation

on

B

.

The

definition

generalizes readily to a family of transformations with
common domain

T

i

:

A

→

A

,

i

∈

I

In this case we say that a subset

B

⊂

A

is fixed, if it is fixed
by all the elements of the family, i.e. whenever

B

⊂

⋂

i

∈

I

A

T

i

.

Title

fix (transformation action)

Canonical

name

FixtransformationAction

Date of creation

2013-03-22 12:26:12

Last modified on

2013-03-22 12:26:12

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

15

Author

rmilson (146)

Entry type

Definition

Classification

msc 03E20

Synonym

fix

Synonym

fixed

Synonym

fixes

Related topic

Invariant

Related topic

Transformation

Related topic

Fix2

Defines

fixed set

Generated on Thu Feb 8 20:12:52 2018 by

LaTeXML
functional

functional

Definition

A

functional

T

is a

function

mapping a

function space

(often a

vector space

)

V

in to a field of scalars

K

, typically taken to be

ℝ

or

ℂ

.

Discussion

Examples of functionals include the

integral

and entropy. A functional

f

is often indicated by the use of brackets,

T

⁢

[

x

]

rather than

T

⁢

(

x

)

.

The

linear functionals

are those functionals

T

that

•

T

⁢

(

x

+

y

)

=

T

⁢

(

x

)

+

T

⁢

(

y

)

•

T

⁢

(

c

⁢

x

)

=

c

⁢

T

⁢

(

x

)

for any

c

∈

K

,

x

,

y

∈

V

.

Title

functional

Canonical name

Functional

Date of creation

2013-03-22 12:29:45

Last modified on

2013-03-22 12:29:45

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Definition

Classification

msc 03E20

Defines

linear functional

Generated on Thu Feb 8 20:12:56 2018 by

LaTeXML
generalized Cartesian product

generalized Cartesian product

Given any family of sets

{

A

j

}

j

∈

J

indexed by

an

index set

J

, the

generalized Cartesian product

∏

j

∈

J

A

j

is the set of all

functions

f

:

J

→

⋃

j

∈

J

A

j

such that

f

⁢

(

j

)

∈

A

j

for all

j

∈

J

.

For each

i

∈

J

, the

projection map

π

i

:

∏

j

∈

J

A

j

→

A

i

is the function defined by

π

i

⁢

(

f

)

:=

f

⁢

(

i

)

.

The generalized Cartesian product is the

product

(

http://planetmath.org/CategoricalDirectProduct

) in the

category of sets

.

The

axiom of choice

is the statement that the generalized Cartesian product of

nonempty sets

is nonempty.
The generalized Cartesian product is usually called the

Cartesian product

.

Title

generalized Cartesian product

Canonical name

GeneralizedCartesianProduct

Date of creation

2013-03-22 11:49:02

Last modified on

2013-03-22 11:49:02

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

15

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 03E20

Related topic

CartesianProduct

Related topic

ProductTopology

Related topic

AxiomOfChoice

Related topic

OrderedTuplet

Related topic

FunctorCategory2

Defines

projection map

Generated on Thu Feb 8 20:12:58 2018 by

LaTeXML
graph

graph

The graph of a

function

f

:

X

→

Y

is the subset of

X

×

Y

given by

{

(

x

,

f

⁢

(

x

)

)

:

x

∈

X

}

.

Title

graph

Canonical name

Graph1

Date of creation

2013-03-22 13:13:53

Last modified on

2013-03-22 13:13:53

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

12

Author

Koro (127)

Entry type

Definition

Classification

msc 03E20

Synonym

function graph

Related topic

ClosedGraphTheorem

Generated on Thu Feb 8 20:13:00 2018 by

LaTeXML
identity map

identity map

Definition

If

X

is a set, then the

identity map

in

X

is the mapping
that maps each element in

X

to itself.

0.0.1

Properties

1.

An identity map is always a

bijection

.

2.

Suppose

X

has two

topologies

τ

1

and

τ

2

. Then
the identity mapping

I

:

(

X

,

τ

1

)

→

(

X

,

τ

2

)

is

continuous

if and only if

τ

1

is finer than

τ

2

, i.e.,

τ

1

⊂

τ

2

.

3.

The identity map on the

n

-sphere, is
homotopic (

http://planetmath.org/HomotopyOfMaps

)
to the

antipodal map

A

:

S

n

→

S

n

if

n

is odd

[

1

]

.

References

1

V. Guillemin, A. Pollack,

Differential topology

, Prentice-Hall Inc., 1974.

Title

identity map

Canonical

name

IdentityMap

Date of creation

2013-03-22 14:03:43

Last modified on

2013-03-22 14:03:43

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

7

Author

bwebste (988)

Entry type

Definition

Classification

msc 03E20

Synonym

identity mapping

Synonym

identity operator

Synonym

identity function

Related topic

ZeroMap

Related topic

IdentityMatrix

Generated on Thu Feb 8 20:13:02 2018 by

LaTeXML
inclusion mapping

inclusion mapping

Definition

Let

X

be a subset of

Y

. Then the

inclusion map

from

X

to

Y

is the

mapping

ι

:

X

→

Y

x

↦

x

.

In other words, the inclusion map is simply a fancy way to say
that every

element

in

X

is also an element in

Y

.

To indicate that a mapping is an inclusion mapping, one usually writes

↪

instead of

→

when defining or mentioning an
inclusion map. This hooked arrow symbol

↪

can be
seen as

combination

of the symbols

⊂

and

→

.
In the above definition, we have not used this convention.
However, examples of this convention would be:

•

Let

ι

:

X

↪

Y

be the inclusion map from

X

to

Y

.

•

We have the inclusion

S

n

↪

ℝ

n

+

1

.

Title

inclusion mapping

Canonical

name

InclusionMapping

Date of creation

2013-03-22 13:43:08

Last modified on

2013-03-22 13:43:08

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

9

Author

Koro (127)

Entry type

Definition

Classification

msc 03E20

Synonym

inclusion map

Synonym

inclusion

Related topic

Pullback2

Generated on Thu Feb 8 20:13:04 2018 by

LaTeXML
inductive set

inductive set

An

inductive set

is a set

X

with the

property

that, for every

x

∈

X

, the

successor

x

′

of

x

is also an element of

X

.

One major example of an inductive set is the set of natural numbers

ℕ

.

Title

inductive set

Canonical

name

InductiveSet

Date of creation

2013-03-22 11:50:01

Last modified on

2013-03-22 11:50:01

Owner

djao (24)

Last modified by

djao (24)

Numerical id

12

Author

djao (24)

Entry type

Definition

Classification

msc 03E20

Classification

msc 45D05

Synonym

successor set

Related topic

NaturalNumber

Generated on Thu Feb 8 20:13:06 2018 by

LaTeXML