aarjaneiro/planetmath huggingface dataset:
idempotency

idempotency

If

(

S

,

*

)

is a magma, then an

element

x

∈

S

is said to be

idempotent

if

x

*

x

=

x

.
For example, every

identity element

is idempotent, and in a group this is the only idempotent element.
An idempotent element is often just called an idempotent.

If every element of the magma

(

S

,

*

)

is idempotent, then the

binary operation

*

(or the magma itself) is said to be idempotent. For example, the

∧

and

∨

operations

in a

lattice

(

http://planetmath.org/Lattice

) are idempotent, because

x

∧

x

=

x

and

x

∨

x

=

x

for all

x

in the lattice.

A

function

f

:

D

→

D

is said to be idempotent if

f

∘

f

=

f

. (This is just a special case of the first definition above, the magma in question being

(

D

D

,

∘

)

, the monoid of all functions from

D

to

D

with the operation of function composition.) In other words,

f

is idempotent if and only if repeated application of

f

has the same effect as a single application:

f

⁢

(

f

⁢

(

x

)

)

=

f

⁢

(

x

)

for all

x

∈

D

. An idempotent

linear transformation

from a

vector space

to itself is called a

projection

.

Title

idempotency

Canonical name

Idempotency

Date of creation

2013-03-22 12:27:31

Last modified on

2013-03-22 12:27:31

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

21

Author

yark (2760)

Entry type

Definition

Classification

msc 20N02

Related topic

BooleanRing

Related topic

PeriodOfMapping

Related topic

Idempotent2

Defines

idempotent

Generated on Fri Feb 9 19:37:03 2018 by

LaTeXML
left identity and right identity

left identity and right identity

Let

G

be a

groupoid

. An element

e

∈

G

is called a

left identity

element

if

e

⁢

x

=

x

for all

x

∈

G

. Similarly,

e

is a

right identity element

if

x

⁢

e

=

x

for all

x

∈

G

.

An element which is both a left and a right identity is an

identity element

.

A groupoid may have more than one left identify element: in fact the operation
defined by

x

⁢

y

=

y

for all

x

,

y

∈

G

defines a groupoid (in fact, a

semigroup

) on any set

G

, and every element is a left identity.

But as soon as a groupoid has both a left and a right identity, they are necessarily unique and equal. For if

e

is a left identity and

f

is a right identity, then

f

=

e

⁢

f

=

e

.

Title

left identity and right identity

Canonical name

LeftIdentityAndRightIdentity

Date of creation

2013-03-22 13:02:05

Last modified on

2013-03-22 13:02:05

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

5

Author

mclase (549)

Entry type

Definition

Classification

msc 20N02

Classification

msc 20M99

Related topic

IdentityElement

Related topic

Unity

Defines

left identity

Defines

right identity

Generated on Fri Feb 9 19:37:15 2018 by

LaTeXML
zero elements

zero elements

Let

S

be a

semigroup

. An

element

z

is called a

right zero

[resp.

left zero

] if

x

⁢

z

=

z

[resp.

z

⁢

x

=

z

] for all

x

∈

S

.

An element which is both a left and a right zero is called a

zero element

.

A semigroup may have many left zeros or right zeros, but if it has at least one of each, then they are necessarily equal, giving a unique (two-sided) zero element.

More generally, these

definitions

and statements are valid for a

groupoid

.

It is customary to use the symbol

θ

for the zero element of a semigroup.

Proposition 1

.

If a groupoid has a left zero

0

L

and a right zero

0

R

, then

0

L

=

0

R

.

Proof.

0

L

=

0

L

⁢

0

R

=

0

R

. ∎

Proposition 2

.

If

0

is a left zero in a semigroup

S

, then so is

x

⁢

0

for every

x

∈

S

.

Proof.

For any

y

∈

S

,

(

x

⁢

0

)

⁢

y

=

x

⁢

(

0

⁢

y

)

=

x

⁢

0

. As a result,

x

⁢

0

is a left zero of

S

.
∎

Proposition 3

.

If

0

is the unique left zero in a semigroup

S

, then it is also the zero element.

Proof.

By

assumption

and the previous

proposition

,

x

⁢

0

is a left zero for every

x

∈

S

. But

0

is the unique left zero in

S

, we must have

x

⁢

0

=

0

, which means that

0

is a right zero element, and hence a zero element by the first proposition.
∎

Title

zero elements

Canonical

name

ZeroElements

Date of creation

2013-03-22 13:02:19

Last modified on

2013-03-22 13:02:19

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

6

Author

mclase (549)

Entry type

Definition

Classification

msc 20N02

Classification

msc 20M99

Related topic

Semigroup

Related topic

NullSemigroup

Related topic

AbsorbingElement

Defines

zero

Defines

zero element

Defines

right zero

Defines

left zero

Generated on Fri Feb 9 19:37:16 2018 by

LaTeXML
alternative definition of a quasigroup

alternative definition of a quasigroup

In the parent entry, a

quasigroup

is defined as a set, together with a

binary operation

on it satisfying two

formulas

, both of which using

existential quantifiers

. In this entry, we give an alternative, but

equivalent

,

definition

of a quasigroup using only universally quantified formulas. In other words, the class of quasigroups is an

equational class

.

Definition

. A

quasigroup

is a set

Q

with three binary operations

⋅

(

multiplication

),

\

(

left division

), and

/

(

right division

), such that the following are satisfied:

•

(

Q

,

⋅

)

is a

groupoid

(not in the category theoretic sense)

•

(left division

identities

) for all

a

,

b

∈

Q

,

a

\

(

a

⋅

b

)

=

b

and

a

⋅

(

a

\

b

)

=

b

•

(right division identities) for all

a

,

b

∈

Q

,

(

a

⋅

b

)

/

b

=

a

and

(

a

/

b

)

⋅

b

=

a

Proposition 1

.

The two definitions of a quasigroup are equivalent.

Proof.

Suppose

Q

is a quasigroup using the definition given in the parent entry (

http://planetmath.org/LoopAndQuasigroup

). Define

\

on

Q

as follows: for

a

,

b

∈

Q

, set

a

\

b

:=

c

where

c

is the unique

element

such that

a

⋅

c

=

b

. Because

c

is unique,

\

is

well-defined

. Now, let

x

=

a

⋅

b

and

y

=

a

\

x

. Since

a

⋅

y

=

x

=

a

⋅

b

, and

y

is uniquely determined, this forces

y

=

b

. Next, let

x

=

a

\

b

, then

a

⋅

x

=

b

, or

a

⋅

(

a

\

b

)

=

b

. Similarly, define

/

on

Q

so that

a

/

b

is the unique element

d

such that

d

⋅

b

=

a

. The verification of the two right division identities is left for the reader.

Conversely

, let

Q

be a quasigroup as defined in this entry. For any

a

,

b

∈

Q

, let

c

=

a

\

b

and

d

=

b

/

a

. Then

a

⋅

c

=

a

⋅

(

a

\

b

)

=

b

and

d

⋅

a

=

(

b

/

a

)

⋅

a

=

b

.
∎

Title

alternative definition of a quasigroup

Canonical

name

AlternativeDefinitionOfAQuasigroup

Date of creation

2013-03-22 18:28:56

Last modified on

2013-03-22 18:28:56

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 20N05

Related topic

Supercategory

Defines

left division

Defines

right division

Generated on Fri Feb 9 19:37:25 2018 by

LaTeXML
an associative quasigroup is a group

an associative quasigroup is a group

Proposition 1

.

Let

G

be a set and

⋅

a

binary operation

on

G

. Write

a

⁢

b

for

a

⋅

b

. The following are

equivalent

:

1.

(

G

,

⋅

)

is an

associative

quasigroup

.

2.

(

G

,

⋅

)

is an associative loop.

3.

(

G

,

⋅

)

is a group.

Proof.

We will prove this in the following direction

(

1

)

⇒

(

2

)

⇒

(

3

)

⇒

(

1

)

.

(

1

)

⇒

(

2

)

.

Let

x

∈

G

, and

e

1

,

e

2

∈

G

such that

x

⁢

e

1

=

x

=

e

2

⁢

x

. So

x

⁢

e

1

2

=

x

⁢

e

1

=

x

, which shows that

e

1

2

=

e

1

. Let

a

∈

G

be such that

e

1

⁢

a

=

x

. Then

e

2

⁢

e

1

⁢

a

=

e

2

⁢

x

=

x

=

e

1

⁢

a

, so that

e

2

⁢

e

1

=

e

1

=

e

1

2

, or

e

2

=

e

1

. Set

e

=

e

1

. For any

y

∈

G

, we have

e

⁢

y

=

e

2

⁢

y

, so

y

=

e

⁢

y

. Similarly,

y

⁢

e

=

y

⁢

e

2

implies

y

=

y

⁢

e

. This shows that

e

is an

identity

of

G

.

(

2

)

⇒

(

3

)

.

First note that all of the group axioms are automatically satisfied in

G

under

⋅

, except the

existence

of an (two-sided) inverse element, which we are going to verify presently. For every

x

∈

G

, there are unique

elements

y

and

z

such that

x

⁢

y

=

z

⁢

x

=

e

. Then

y

=

e

⁢

y

=

(

z

⁢

x

)

⁢

y

=

z

⁢

(

x

⁢

y

)

=

z

⁢

e

=

z

. This shows that

x

has a unique two-sided

inverse

x

-

1

:=

y

=

z

. Therefore,

G

is a group under

⋅

.

(

3

)

⇒

(

1

)

.

Every group is clearly a quasigroup, and the binary operation is associative.

This

completes

the proof.
∎

Remark

. In fact, if

⋅

on

G

is flexible, then every element in

G

has a unique inverse: for

z

⁢

(

x

⁢

z

)

=

(

z

⁢

x

)

⁢

z

=

e

⁢

z

=

z

=

z

⁢

e

, so by

left division

(by

z

), we get

x

⁢

z

=

e

=

x

⁢

y

, and therefore

z

=

y

, again by left division (by

x

). However,

G

may no longer be a group, because associativity may longer hold.

Title

an associative quasigroup is a group

Canonical name

AnAssociativeQuasigroupIsAGroup

Date of creation

2013-03-22 18:28:50

Last modified on

2013-03-22 18:28:50

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Derivation

Classification

msc 20N05

Related topic

Group

Generated on Fri Feb 9 19:37:34 2018 by

LaTeXML
loop and quasigroup

loop and quasigroup

A

quasigroup

is a

groupoid

G

with the

property

that for every

x

,

y

∈

G

, there are unique elements

w

,

z

∈

G

such that

x

⁢

w

=

y

and

z

⁢

x

=

y

.

A

loop

is a

quasigroup

which has an

identity element

.

What distinguishes a loop from a group is that the former need not satisfy the

associative

law.

Title

loop and quasigroup

Canonical

name

LoopAndQuasigroup

Date of creation

2013-03-22 13:02:08

Last modified on

2013-03-22 13:02:08

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

4

Author

mclase (549)

Entry type

Definition

Classification

msc 20N05

Related topic

Groupoid

Related topic

LoopOfAGraph

Related topic

AlternativeDefinitionOfGroup

Defines

loop

Defines

quasigroup

Generated on Fri Feb 9 19:37:36 2018 by

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medial quasigroup

medial quasigroup

A

medial quasigroup

is a

quasigroup

such that, for any choice of four

elements

a

,

b

,

c

,

d

, one has

(

a

⋅

b

)

⋅

(

c

⋅

d

)

=

(

a

⋅

c

)

⋅

(

b

⋅

d

)

.

Any

commutative

quasigroup is trivially a medial quasigroup. A nontrivial class of examples may be constructed as follows. Take a commutative group

(

G

,

+

)

and two

automorphisms

f

,

g

:

G

→

G

which commute with each other, and an element

c

of

G

. Then, if we define an

operation

⋅

:

G

×

G

→

G

as

x

⋅

y

=

f

⁢

(

a

)

+

g

⁢

(

b

)

+

c

,

(

G

,

⋅

)

is a medial quasigroup.

Reference:

V. D. Belousov, Fundamentals of the theory of quasigroups and loops (in Russian)

Title

medial quasigroup

Canonical name

MedialQuasigroup

Date of creation

2013-03-22 16:27:33

Last modified on

2013-03-22 16:27:33

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

5

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 20N05

Generated on Fri Feb 9 19:37:37 2018 by

LaTeXML
Moufang loop

Moufang loop

Proposition

:

Let

Q

be a nonempty

quasigroup

.

I) The following conditions are

equivalent

.

(

x

⁢

(

y

⁢

z

)

)

⁢

x

=

(

x

⁢

y

)

⁢

(

z

⁢

x

)

for all

⁢

x

,

y

,

z

∈

Q

(1)

(

(

x

⁢

y

)

⁢

z

)

⁢

y

=

x

⁢

(

y

⁢

(

z

⁢

y

)

)

for all

⁢

x

,

y

,

z

∈

Q

(2)

(

x

⁢

z

)

⁢

(

y

⁢

x

)

=

x

⁢

(

(

z

⁢

y

)

⁢

x

)

for all

⁢

x

,

y

,

z

∈

Q

(3)

(

(

y

⁢

z

)

⁢

y

)

⁢

x

=

y

⁢

(

z

⁢

(

y

⁢

x

)

)

for all

⁢

x

,

y

,

z

∈

Q

(4)

II) If

Q

satisfies

those conditions, then

Q

has an

identity element

(i.e.,

Q

is a loop).

For a proof, we refer the reader to the two

references

.
Kunen in [1] shows that that any of the four conditions implies the

existence

of an identity element. And Bol and Bruck [2] show that
the four conditions are equivalent for loops.

Definition

:

A nonempty quasigroup satisfying the conditions
(1)–(4) is called a Moufang quasigroup or, equivalently, a

Moufang
loop

(after Ruth Moufang, 1905–1977).

The 16-element set of unit octonions over

ℤ

is an example
of a nonassociative Moufang loop.
Other examples appear in projective geometry, coding theory, and elsewhere.

References

[1] Kenneth Kunen,

Moufang Quasigroups

, J.

Algebra

83 (1996) 231–234.
(A preprint in PostScript format is available from Kunen’s website:

http://www.math.wisc.edu/ kunen/moufang.ps

Moufang Quasigroups.)

[2] R. H. Bruck,

A Survey of Binary Systems,

Springer-Verlag, 1958.

Title

Moufang loop

Canonical

name

MoufangLoop

Date of creation

2013-03-22 13:50:29

Last modified on

2013-03-22 13:50:29

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

12

Author

yark (2760)

Entry type

Definition

Classification

msc 20N05

Generated on Fri Feb 9 19:37:39 2018 by

LaTeXML
proof of example of medial quasigroup

proof of example of medial quasigroup

We shall proceed by first showing that the

algebraic systems

defined in the parent entry (

http://planetmath.org/MedialQuasigroup

) are

quasigroups

and then showing that the medial

property

is satisfied.

To show that the system is a quasigroup, we need to check the solubility of

equations

. Let

x

and

y

be two elements of

G

. Then, by

definition

of

⋅

, the equation

x

⋅

z

=

y

is

equivalent

to

f

⁢

(

x

)

+

g

⁢

(

z

)

+

c

=

y

.

This is equivalent to

g

⁢

(

z

)

=

y

-

c

-

f

⁢

(

x

)

.

Since

g

is an

automorphism

, there will exist a unique solution

z

to this equation.

Likewise, the equation

z

⋅

x

=

y

is equivalent to

f

⁢

(

z

)

+

g

⁢

(

x

)

+

c

=

y

which, in turn is equivalent to

f

⁢

(

z

)

=

y

-

c

-

g

⁢

(

x

)

,

so we may also find a unique

z

such that

z

⋅

x

=

y

. Hence,

(

G

,

⋅

)

is a quasigroup.

To check the medial property, we use the definition of

⋅

to conclude that

(

x

⋅

y

)

⋅

(

z

⋅

w

)

=

(

f

⁢

(

x

)

+

g

⁢

(

y

)

+

c

)

⋅

(

f

⁢

(

z

)

+

g

⁢

(

w

)

+

c

)

=

f

⁢

(

f

⁢

(

x

)

+

g

⁢

(

y

)

+

c

)

+

g

⁢

(

f

⁢

(

z

)

+

g

⁢

(

w

)

+

c

)

+

c

Since

f

and

g

are automorphisms and the group is

commutative

, this equals

f

⁢

(

f

⁢

(

x

)

)

+

f

⁢

(

g

⁢

(

y

)

)

+

g

⁢

(

f

⁢

(

z

)

)

+

g

⁢

(

g

⁢

(

w

)

)

+

f

⁢

(

c

)

+

g

⁢

(

c

)

+

c

.

Since

f

and

g

commute this, in turn, equals

f

⁢

(

f

⁢

(

x

)

)

+

g

⁢

(

f

⁢

(

y

)

)

+

f

⁢

(

g

⁢

(

z

)

)

+

g

⁢

(

g

⁢

(

w

)

)

+

f

⁢

(

c

)

+

g

⁢

(

c

)

+

c

.

Using the commutative and

associative

laws, we may regroup this

expression

as follows:

(

f

⁢

(

f

⁢

(

x

)

)

+

f

⁢

(

g

⁢

(

z

)

)

+

f

⁢

(

c

)

)

+

(

g

⁢

(

f

⁢

(

y

)

)

+

g

⁢

(

g

⁢

(

w

)

)

+

g

⁢

(

c

)

)

+

c

Because

f

and

g

are automorphisms, this equals

f

⁢

(

f

⁢

(

x

)

+

g

⁢

(

z

)

+

c

)

+

g

⁢

(

f

⁢

(

y

)

+

g

⁢

(

w

)

+

c

)

+

c

By defintion of

⋅

, this equals

f

⁢

(

x

⋅

z

)

+

g

⁢

(

y

⋅

z

)

+

c

,

which equals

(

x

⋅

z

)

⋅

(

y

⋅

z

)

, so we have

(

x

⋅

y

)

⋅

(

z

⋅

w

)

=

(

x

⋅

z

)

⋅

(

y

⋅

z

)

.

Thus, the medial property is satisfied, so we have a

medial quasigroup

.

Title

proof of example of medial quasigroup

Canonical

name

ProofOfExampleOfMedialQuasigroup

Date of creation

2013-03-22 16:27:35

Last modified on

2013-03-22 16:27:35

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

8

Author

rspuzio (6075)

Entry type

Proof

Classification

msc 20N05

Generated on Fri Feb 9 19:37:42 2018 by

LaTeXML
mixed group

mixed group

A

mixed group

is a

partial groupoid

G

such that

G

contains a non-empty subset

K

, called the

kernel

of

G

, with the following conditions:

1.

if

a

,

b

∈

G

, then

a

⁢

b

is defined iff

a

∈

K

,

2.

if

a

,

b

∈

K

and

c

∈

G

, then

(

a

⁢

b

)

⁢

c

=

a

⁢

(

b

⁢

c

)

,

3.

if

a

∈

K

, then

K

⊆

a

⁢

K

∩

K

⁢

a

,

4.

if

a

∈

K

and

b

∈

G

such that

a

⁢

b

=

b

, then

a

⁢

c

=

c

for all

c

∈

G

.

Mixed groups are

generalizations

of groups, as the following

proposition

illustrates:

Proposition 1

.

If

K

=

G

, then

G

is a group.

Proof.

G

is a

groupoid

by condition 1, and a

semigroup

by condition 2.

Now, by condition 3, given

a

∈

G

, there is

b

∈

G

such that

b

⁢

a

=

a

, so that

b

⁢

c

=

c

for all

c

∈

G

by condition 4. In other words,

b

is a

left identity

of

G

. Again, by condition 3, for every

a

∈

G

, there is a

d

∈

G

such that

b

=

d

⁢

a

. So

a

⁢

d

=

a

⁢

(

b

⁢

d

)

=

a

⁢

(

d

⁢

a

)

⁢

d

=

(

a

⁢

d

)

2

, so, by condition 4,

a

⁢

d

⁢

x

=

x

for all

x

∈

G

. In particular, set

x

=

a

, we get

a

=

(

a

⁢

d

)

⁢

a

=

a

⁢

(

d

⁢

a

)

=

a

⁢

b

. Hence,

b

is a two-sided

identity

, and

G

is a monoid.

Finally, by condition 3, for every

a

∈

G

, there are

c

,

d

∈

G

, such that

b

=

a

⁢

c

=

d

⁢

a

. So,

c

=

b

⁢

c

=

(

d

⁢

a

)

⁢

c

=

d

⁢

(

a

⁢

c

)

=

d

⁢

b

=

d

, showing that

a

has a two-sided

inverse

. This means that

G

is a group.
∎

For a non-trivial example of a mixed group, let

G

be a group and

H

a

subgroup

of

G

. Define a new

multiplication

⋅

on

G

as follows:

a

⋅

b

is defined iff

a

∈

H

, and if

a

⋅

b

is defined, it is defined as

a

⁢

b

, the group multiplication of

a

and

b

. Then

(

G

,

⋅

)

is a mixed group. Clearly,

associativity

of

⋅

is automatically satisfied. Next, pick any

a

∈

H

, then, for any

b

∈

H

,

a

-

1

⋅

b

and

b

⋅

a

-

1

are both elements of

H

, so that

b

∈

a

⋅

H

∩

H

⋅

a

, and condition 3 is also satisfied. Finally, if

a

∈

H

and

b

∈

G

such that

a

⋅

b

=

b

, then

a

is the

multiplicative identity

of

G

, clearly

a

⋅

c

=

c

for all

c

∈

G

.

References

1

R. H. Bruck,

A Survey of Binary Systems

, Springer-Verlag, 1966

2

R. Baer,

Zur Einordnung der Theorie der Mischgruppen in die Gruppentheorie

, S.-B. Heidelberg. Akad. Wiss., Math.-naturwiss. KI. 1928, 4, 13 pp

3

R. Baer,

Über die Zerlegungen einer Mischgruppe nach einer Untermischgruppe

, S.-B. Heidelberg. Akad. Wiss., Math.-naturwiss. KI. 1928, 5, 13 pp

4

A. Loewy,

Über abstrakt definierte Transmutationssysteme oder Mischgruppen

, J. reine angew. Math. 157, pp 239-254, 1927

Title

mixed group

Canonical

name

MixedGroup

Date of creation

2013-03-22 18:42:32

Last modified on

2013-03-22 18:42:32

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 20N99

Defines

kernel

Generated on Fri Feb 9 19:38:24 2018 by

LaTeXML
fixed-point subspace

fixed-point subspace

Let

Σ

⊂

Γ

be a

subgroup

where

Γ

is a

compact Lie Group

acting on a

vector space

V

. The

fixed-point subspace

of

Σ

is

Fix

⁡

(

Σ

)

=

{

x

∈

V

∣

σ

⁢

x

=

x

,

∀

σ

∈

Σ

}

Fix

⁡

(

Σ

)

is a

linear subspace

of

V

since

Fix

⁡

(

Σ

)

=

⋂

σ

∈

Σ

ker

⁡

(

σ

-

I

)

where

I

is the

identity

. If it is important to specify the space

V

we use the following notation

Fix

V

⁡

(

Σ

)

.

References

GSS

Golubitsky, Martin. Stewart, Ian. Schaeffer, G. David: Singularities and Groups in Bifurcation Theory

(Volume II)

. Springer-Verlag, New York, 1988.

Title

fixed-point subspace

Canonical

name

FixedpointSubspace

Date of creation

2013-03-22 13:44:31

Last modified on

2013-03-22 13:44:31

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 22-00

Classification

msc 15A03

Generated on Fri Feb 9 19:38:44 2018 by

LaTeXML
Sorgenfrey half-open plane

Sorgenfrey half-open plane

The

Sorgenfrey plane

is the

product

of the

Sorgenfrey line

with itself. This

topology

can also be described as the topology on

ℝ

2

which arises from the basis

{

[

a

,

b

)

×

[

c

,

d

)

∣

a

,

b

,

c

,

d

∈

ℝ

,

a

<

b

,

c

<

d

}

.

It is interesting to note that, even though the Sorgenfrey line
enjoys the Lindelöf

property

(

http://

planetmath

.org/lindelofspace

), the
Sorgenfrey plane does not. To see this, one can note that the
line

x

+

y

=

0

is a

closed subset

in this topology and that
the induced topology on this line is the

discrete topology

.
Since the Lindelöf property is weakly hereditary, the
discrete topology on the real line would have to be Lindelöf
if the Sorgenfrey plane topology were Lindelöf. However, the
discrete topology on an uncountable set can never have the
Lindelöf property, so the Sorgenfrey topology cannot have
this property either.

Reference

Sorgenfrey, R. H.

On the Topological Product of

Paracompact Spaces

, Bulletin of the

American Mathematical Society

, (1947) 631-632

Title

Sorgenfrey half-open plane

Canonical

name

SorgenfreyHalfopenPlane

Date of creation

2014-11-06 13:51:15

Last modified on

2014-11-06 13:51:15

Owner

rspuzio (6075)

Last modified by

pahio (2872)

Numerical id

7

Author

rspuzio (2872)

Entry type

Definition

Classification

msc 22-00

Classification

msc 55-00

Classification

msc 54-00

Synonym

Sorgenfrey’s half-open square topology

Synonym

Sorgenfrey plane

Generated on Fri Feb 9 19:38:46 2018 by

LaTeXML
topological properties of the Sorgenfrey half-open plane

topological properties of the Sorgenfrey half-open plane

The

Sorgenfrey plane

has the following properties:

•

Hausdorff

•

completely regular

•

separable

The Sorgenfrey plane does not have the following properties:

•

Lindelof

•

normal

•

second countable

Title

topological properties of the Sorgenfrey half-open plane

Canonical name

TopologicalPropertiesOfTheSorgenfreyHalfopenPlane

Date of creation

2013-03-22 14:48:21

Last modified on

2013-03-22 14:48:21

Owner

drini (3)

Last modified by

drini (3)

Numerical id

7

Author

drini (3)

Entry type

Definition

Classification

msc 22-00

Classification

msc 55-00

Classification

msc 54-00

Generated on Fri Feb 9 19:38:47 2018 by

LaTeXML