aarjaneiro/planetmath huggingface dataset:
triangle inequality

triangle inequality

Let

(

X

,

d

)

be a

metric space

. The

triangle inequality

states
that for any three points

x

,

y

,

z

∈

X

we have

d

⁢

(

x

,

y

)

≤

d

⁢

(

x

,

z

)

+

d

⁢

(

z

,

y

)

.

The name comes from the special case of

ℝ

n

with the

standard
topology

, and geometrically meaning that in any

triangle

, the sum of
the lengths of two sides is

greater

(or equal) than the third.

Actually, the triangle inequality is one of the properties that define
a metric, so it holds in any metric space. Two important cases are

ℝ

with

d

⁢

(

x

,

y

)

=

|

x

-

y

|

and

ℂ

with

d

⁢

(

x

,

y

)

=

∥

x

-

y

∥

(here
we are using

complex modulus

, not

absolute value

).

There is a second triangle inequality, sometimes called the

reverse triangle inequality

, which also holds in any metric
space and is derived from the definition of metric:

d

⁢

(

x

,

y

)

≥

|

d

⁢

(

x

,

z

)

-

d

⁢

(

z

,

y

)

|

.

In

Euclidean geometry

, this

inequality

is expressed by saying that each
side of a triangle is greater than the difference of the other two.

The reverse triangle inequality can be proved from the first triangle
inequality, as we now show.

Let

x

,

y

,

z

∈

X

be given. For any

a

,

b

,

c

∈

X

, from the first
triangle inequality we have:

d

⁢

(

a

,

b

)

≤

d

⁢

(

a

,

c

)

+

d

⁢

(

c

,

b

)

and thus (using

d

⁢

(

b

,

c

)

=

d

⁢

(

c

,

b

)

for any

b

,

c

∈

X

):

d

⁢

(

a

,

c

)

≥

d

⁢

(

a

,

b

)

-

d

⁢

(

b

,

c

)

(1)

and writing (

1

) with

a

=

x

,

b

=

z

,

c

=

y

:

d

⁢

(

x

,

y

)

≥

d

⁢

(

x

,

z

)

-

d

⁢

(

z

,

y

)

(2)

while writing (

1

) with

a

=

y

,

b

=

z

,

c

=

x

we get:

d

⁢

(

y

,

x

)

≥

d

⁢

(

y

,

z

)

-

d

⁢

(

z

,

x

)

or

d

⁢

(

x

,

y

)

≥

d

⁢

(

z

,

y

)

-

d

⁢

(

x

,

z

)

;

(3)

from (

2

) and (

3

), using the properties of the
absolute value, it follows finally:

d

⁢

(

x

,

y

)

≥

|

d

⁢

(

x

,

z

)

-

d

⁢

(

z

,

y

)

|

which is the second triangle inequality.

Title

triangle inequality

Canonical name

TriangleInequality

Date of creation

2013-03-22 12:14:49

Last modified on

2013-03-22 12:14:49

Owner

drini (3)

Last modified by

drini (3)

Numerical id

12

Author

drini (3)

Entry type

Definition

Classification

msc 54-00

Classification

msc 54E35

Related topic

ProofOfLimitRuleOfProduct

Related topic

TriangleInequalityOfComplexNumbers

Defines

reverse triangle inequality

Generated on Sat Feb 10 11:12:03 2018 by

LaTeXML
universal covering space

universal covering space

Let

X

be a

topological space

. A

universal covering space

is a covering space

X

~

of

X

which is

connected

and

simply connected

.

If

X

is based, with

basepoint

x

, then a

based cover

of

X

is cover of

X

which is also a based space with a basepoint

x

′

such that the

covering

is a map of based spaces. Note that any cover can be made into a based cover by choosing a basepoint from the pre-images of

x

.

The universal covering space has the following

universal property

: If

π

:

(

X

~

,

x

0

)

→

(

X

,

x

)

is a based

universal cover

, then for any connected based cover

π

′

:

(

X

′

,

x

′

)

→

(

X

,

x

)

, there is a unique covering map

π

′′

:

(

X

~

,

x

0

)

→

(

X

′

,

x

′

)

such that

π

=

π

′

∘

π

′′

.

Clearly, if a universal covering exists, it is unique up to unique

isomorphism

. But not every topological space has a universal cover. In fact

X

has a universal cover if and only if it is semi-locally simply connected (for example, if it is a

locally finite

CW-complex

or a manifold).

Title

universal covering space

Canonical name

UniversalCoveringSpace

Date of creation

2013-03-22 12:15:34

Last modified on

2013-03-22 12:15:34

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

7

Author

bwebste (988)

Entry type

Definition

Classification

msc 54-00

Synonym

universal cover

Related topic

OmegaSpectrum

Generated on Sat Feb 10 11:12:05 2018 by

LaTeXML
when are balls separated

when are balls separated

Let

(

X

,

d

)

be a

metric space

, and let

B

r

⁢

(

x

)

be the

x

-centered
open ball of radius

r

. If

d

⁢

(

x

,

y

)

≥

r

+

s

, then the balls

B

r

⁢

(

x

)

and

B

s

⁢

(

y

)

are

separated

.

To prove this, suppose that

B

r

⁢

(

x

)

and

B

s

⁢

(

y

)

are not separated.
Then there exists a

z

∈

X

such that either

d

⁢

(

x

,

z

)

<

r

,

d

⁢

(

y

,

z

)

≤

s

,

or

d

⁢

(

x

,

z

)

≤

r

,

d

⁢

(

y

,

z

)

<

s

.

In either case,

d

⁢

(

x

,

y

)

≤

d

⁢

(

x

,

z

)

+

d

⁢

(

z

,

y

)

<

r

+

s

.

Title

when are balls separated

Canonical name

WhenAreBallsSeparated

Date of creation

2013-03-22 15:16:40

Last modified on

2013-03-22 15:16:40

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

6

Author

matte (1858)

Entry type

Example

Classification

msc 54-00

Classification

msc 54D05

Generated on Sat Feb 10 11:12:07 2018 by

LaTeXML
zero dimensional

zero dimensional

Definition 1

.

[

1

,

2

]

Suppose

X

is a

topological space

. If

X

has a basis consising of

clopen sets

, then

X

is said to be .

Examples of zero-dimensional spaces are: the set

ℚ

of

rational numbers

(with

subspace topology

induced from the usual

metric topology

on

ℝ

, the set of real numbers), the

Cantor space

, as well as the

Sorgenfrey line

.

The

concepts

of zero-dimentionality and total disconnectedness are closely related. Indeed, every zero-dimentional

T

1

space (

http://

planetmath

.org/T1Space

) is

totally disconnected

. Furthermore, if a topological space is

locally compact

and

Hausdorff

, then the notions of zero-dimentionality and total disconnectedness are

equivalent

.

References

1

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

Counterexamples in topology

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

2

S. Willard,

General Topology

,
Addison-Wesley, Publishing Company, 1970.

Title

zero dimensional

Canonical

name

ZeroDimensional

Date of creation

2013-03-22 14:41:05

Last modified on

2013-03-22 14:41:05

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

9

Author

matte (1858)

Entry type

Definition

Classification

msc 54-00

Synonym

zero-dimensional

Related topic

SeparationAxioms

Generated on Sat Feb 10 11:12:09 2018 by

LaTeXML
Alexandroff space

Alexandroff space

Topological space

X

is called

Alexandroff

if the

intersection

of every family of

open sets

is open.

Of course every finite topological space is Alexandroff, but there are also bigger Alexandroff spaces. For example let

ℝ

denote the set of real numbers and let

τ

=

{

[

a

,

∞

)

|

a

∈

ℝ

}

∪

{

(

b

,

∞

)

|

b

∈

ℝ

}

. Then

τ

is a

topology

on

ℝ

and

(

ℝ

,

τ

)

is an Alexandroff space.

If

X

is an Alexandroff space and

A

⊆

X

, then we may talk about smallest open

neighbourhood

of

A

. Indeed, let

A

o

=

⋂

{

U

⊆

X

|

U

⁢

is open and

⁢

A

⁢

is contained in

⁢

U

}

.

Then

A

o

is open.

Title

Alexandroff space

Canonical

name

AlexandroffSpace

Date of creation

2013-03-22 18:45:41

Last modified on

2013-03-22 18:45:41

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

5

Author

joking (16130)

Entry type

Definition

Classification

msc 54A05

Generated on Sat Feb 10 11:12:11 2018 by

LaTeXML
Alexandroff space is T1 if and only if it is discrete

Alexandroff space is T1 if and only if it is discrete

Proposition

.

Let

X

be an Alexandroff space. Then

X

is

T

1

if and only if

X

is

discrete

.

Proof.

,,

⇐

” It is easy to see, that every

discrete space

is Alexandroff and

T

1

.

,,

⇒

” Recall that

topological space

is

T

1

if and only if every subset is equal to the

intersection

of all its open

neighbourhoods

. So let

x

∈

X

. Then the intersection of all open neighbourhoods

{

x

}

o

of

x

is equal to

{

x

}

. But since

X

is Alexandroff, then

{

x

}

o

=

{

x

}

is open and thus points are open. Therefore

X

is discrete.

□

Title

Alexandroff space is T1 if and only if it is discrete

Canonical

name

AlexandroffSpaceIsT1IfAndOnlyIfItIsDiscrete

Date of creation

2013-03-22 18:46:08

Last modified on

2013-03-22 18:46:08

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

4

Author

joking (16130)

Entry type

Derivation

Classification

msc 54A05

Generated on Sat Feb 10 11:12:12 2018 by

LaTeXML
characterization of connected compact metric spaces.

characterization of connected compact metric spaces.

Let

(

A

,

d

)

be a compact metric space. Then

A

is

connected

⟺

∀

x

,

y

∈

A

,

∀

ε

>

0

∃

n

∈

ℕ

,

and

p

1

,

…

,

p

n

∈

A

,

such that

p

1

=

x

,

p

n

=

y

and

d

(

p

i

,

p

i

+

1

)

<

ε

(

i

=

1

,

…

,

n

-

1

)

Title

characterization

of connected compact metric spaces.

Canonical

name

CharacterizationOfConnectedCompactMetricSpaces

Date of creation

2013-03-22 14:05:37

Last modified on

2013-03-22 14:05:37

Owner

gumau (3545)

Last modified by

gumau (3545)

Numerical id

12

Author

gumau (3545)

Entry type

Theorem

Classification

msc 54A05

Related topic

LadderConnected

Generated on Sat Feb 10 11:12:14 2018 by

LaTeXML
closed point

closed point

Let

X

be a

topological space

and suppose that

x

∈

X

. If

{

x

}

=

{

x

}

¯

then we say that

x

is a

closed point

. In other words,

x

is closed if

{

x

}

is a

closed set

.

For example, the

real line

ℝ

equipped with the usual

metric topology

, every point is a closed point.

More generally, if a topological space is

T

1

(

http://planetmath.org/T1

), then every point in it is closed. If we remove the condition of being

T

1

, then the property fails, as in the case of the

Sierpinski space

X

=

{

x

,

y

}

, whose

open sets

are

∅

,

X

, and

{

x

}

. The

closure

of

{

x

}

is

X

, not

{

x

}

.

Title

closed point

Canonical name

ClosedPoint

Date of creation

2013-03-22 16:22:24

Last modified on

2013-03-22 16:22:24

Owner

jocaps (12118)

Last modified by

jocaps (12118)

Numerical id

9

Author

jocaps (12118)

Entry type

Definition

Classification

msc 54A05

Generated on Sat Feb 10 11:12:16 2018 by

LaTeXML
closure axioms

closure axioms

A

closure operator

on a set

X

is an

operator

which assigns a set

A

c

to each subset

A

of

X

, and such that the following (Kuratowski’s

closure axioms

) hold for any subsets

A

and

B

of

X

:

1.

∅

c

=

∅

;

2.

A

⊂

A

c

;

3.

(

A

c

)

c

=

A

c

;

4.

(

A

∪

B

)

c

=

A

c

∪

B

c

.

The following

theorem

due to Kuratowski says that a closure operator characterizes a unique

topology

on

X

:

Theorem.

Let

c

be a closure operator on

X

, and let

𝒯

=

{

X

-

A

:

A

⊆

X

,

A

c

=

A

}

. Then

𝒯

is a topology on

X

, and

A

c

is the

𝒯

-

closure

of

A

for each subset

A

of

X

.

Title

closure axioms

Canonical name

ClosureAxioms

Date of creation

2013-03-22 13:13:44

Last modified on

2013-03-22 13:13:44

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

9

Author

Koro (127)

Entry type

Definition

Classification

msc 54A05

Synonym

Kuratowski’s closure axioms

Synonym

Kuratowski closure axioms

Related topic

Closure

Defines

closure operator

Generated on Sat Feb 10 11:12:17 2018 by

LaTeXML
closure map

closure map

Let

P

be a poset. A

function

c

:

P

→

P

is called a

closure map

if

•

c

is order preserving,

•

1

P

≤

c

,

•

c

is

idempotent

:

c

∘

c

=

c

.

If the second condition is changed to

c

≤

1

P

, then

c

is called a

dual closure map

on

P

.

For example, the real function

f

such that

f

⁢

(

r

)

is the least integer greater than or equal to

r

is a closure map (see

Archimedean property

). The

rounding

function

[

⋅

]

is an example of a dual closure map.

A

fixed point

of a closure map

c

on

P

is an

element

x

∈

P

such that

c

⁢

(

x

)

=

x

. It is evident that every image point of

c

is a fixed point: for if

x

=

c

⁢

(

a

)

for some

a

∈

P

, then

c

⁢

(

x

)

=

c

⁢

(

c

⁢

(

a

)

)

=

c

⁢

(

a

)

=

x

.

In the example above, any integer is a fixed point of

f

.

Every closure map can be characterized by an interesting decomposition

property

:

c

:

P

→

P

is a closure map iff there is a set

Q

and a residuated function

f

:

P

→

Q

such that

c

=

f

+

∘

f

, where

f

+

denotes the residual of

f

.

Again, in the example above,

f

=

g

+

∘

g

, where

g

:

ℝ

→

ℤ

is the function taking any

real number

r

to the largest integer smaller than

r

.

g

is residuated, and its residual is

g

+

⁢

(

x

)

=

x

+

1

.

Remark

. Closure maps are

generalizations

to

closure operator

on a set (see the parent entry). Indeed, any closure operator on a set

X

takes a subset

A

of

X

to a subset

A

c

of

X

satisfying the closure axioms, where Axiom 2 corresponds to condition 2 above, and Axiom 3 says the

operator

is idempotent. To see that the operator is order preserving, suppose

A

⊆

B

. Then

B

c

=

(

A

∪

B

)

c

=

A

c

∪

B

c

by Axiom 4, and hence

A

c

⊆

B

c

. Axiom 1 says that the

empty set

∅

is a fixed point of the operator. However, in general, this is not the case, for

P

may not even have a

minimal element

, as indicated by the above example.

References

1

T.S. Blyth,

Lattices and Ordered

Algebraic Structures

, Springer, New York (2005).

Title

closure map

Canonical

name

ClosureMap

Date of creation

2013-03-22 18:53:55

Last modified on

2013-03-22 18:53:55

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 54A05

Classification

msc 06A15

Synonym

closure

Synonym

closure function

Synonym

closure operator

Defines

dual closure

Defines

fixed point

Generated on Sat Feb 10 11:12:26 2018 by

LaTeXML
coercive function

coercive function

Definition 1

(coercive function).

Let

X

and

Y

be

topological spaces

.
A function

f

:

X

→

Y

is said to be

coercive

if for every

compact set

J

⊂

Y

there exists a compact set

K

⊂

X

such that

F

⁢

(

X

∖

K

)

⊂

Y

∖

J

.

The general

definition

given above has a clear sense when specialized to the

Euclidean spaces

, as shown in the following result.

Proposition 1

(coercive functions on

R

n

).

A function

f

:

R

n

→

R

m

is coercive if and only if

lim

|

x

|

→

+

∞

⁡

|

f

⁢

(

x

)

|

=

+

∞

.

Title

coercive function

Canonical

name

CoerciveFunction

Date of creation

2013-03-22 15:20:13

Last modified on

2013-03-22 15:20:13

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

5

Author

paolini (1187)

Entry type

Definition

Classification

msc 54A05

Synonym

coercive

Synonym

coercitive

Synonym

coercitive function

Generated on Sat Feb 10 11:12:39 2018 by

LaTeXML
composition with coercive function

composition with coercive function

Theorem 1

.

Suppose

X

,

Y

,

Z

are

topological spaces

,

f

:

X

→

Y

is a

bijective

proper map

, and

g

:

Y

→

Z

is a

coercive

map.
Then

g

∘

f

:

X

→

Z

is a coercive map.

Proof.

Let

J

⊆

Z

be a

compact set

. As

g

is coercive, there
is a compact set

K

⊆

Y

such that

g

⁢

(

Y

∖

K

)

⊆

Z

∖

J

.

Let

I

=

f

-

1

⁢

(

K

)

, and since

f

is a proper map

I

is

compact

.
Thus

(

g

∘

f

)

⁢

(

X

∖

I

)

=

g

⁢

(

Y

∖

K

)

⊆

Z

∖

J

and

g

∘

f

is coercive.
∎

Title

composition with coercive function

Canonical name

CompositionWithCoerciveFunction

Date of creation

2013-03-22 15:20:16

Last modified on

2013-03-22 15:20:16

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

6

Author

matte (1858)

Entry type

Theorem

Classification

msc 54A05

Generated on Sat Feb 10 11:12:41 2018 by

LaTeXML
condensation point

condensation point

Let

X

be a

topological space

and

A

⊂

X

. A point

x

∈

X

is called a

condensation point

of

A

if every open

neighbourhood

of

x

contains

uncountably

many points of

A

.

For example, if

X

=

ℝ

and

A

any subset, then any

accumulation point

of

A

is automatically a condensation point. But if

X

=

ℚ

and

A

any subset, then

A

does not have any condensation points at all.

We have further

classifications

of

condensation point

where the topological space is an

ordered field

. Namely,

1.

unilateral condensation point:

x

is a condensation point of

A

and there is a

positive

ϵ

with either

(

x

-

ϵ

,

x

)

∩

A

countable

or

(

x

,

x

+

ϵ

)

∩

A

countable.

2.

bilateral condensation point:

For all

ϵ

>

0

, we have both

(

x

-

ϵ

,

x

)

∩

A

and

(

x

,

x

+

ϵ

)

∩

A

uncountable

.

If

κ

is any

cardinal

(i.e. an

ordinal

which is the least among all ordinals of the same

cardinality

as itself), then a

κ

-condensation point can be defined similarly.

Title

condensation point

Canonical name

CondensationPoint

Date of creation

2013-03-22 16:40:45

Last modified on

2013-03-22 16:40:45

Owner

sauravbhaumik (15615)

Last modified by

sauravbhaumik (15615)

Numerical id

9

Author

sauravbhaumik (15615)

Entry type

Definition

Classification

msc 54A05

Generated on Sat Feb 10 11:12:42 2018 by

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countable complement topology

countable complement topology

Let

X

be an

infinite set

. We define the

countable complement topology

on

X

by declaring the

empty set

to be open, and a non-empty subset

U

⊂

X

to be open if

X

\

U

is

countable

.

If

X

is countable, then the countable complement topology is just the

discrete topology

, as the

complement

of

any

set is countable and thus open.

Though defined similarly to the finite complement

topology

, the countable complement topology lacks many of the strong

compactness

properties

of the finite complement topology. For example, the countable complement topology on an

uncountable set

gives an example of a

topological space

that is not

weakly countably compact

(but

is

pseudocompact

).

Title

countable complement topology

Canonical

name

CountableComplementTopology

Date of creation

2013-03-22 14:37:56

Last modified on

2013-03-22 14:37:56

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

5

Author

mathcam (2727)

Entry type

Definition

Classification

msc 54A05

Synonym

cocountable topology

Generated on Sat Feb 10 11:12:44 2018 by

LaTeXML
discrete

discrete

This entry aims at highlighting the fact that all uses of the word

discrete

in mathematics are directly related to the core concept of

discrete space

:

•

A

discrete set

is a set that, endowed with the

topology

implied by the context, is a

\PMlinkescaptetext

discrete space. For instance for a subset of

ℝ

n

and without information suggesting otherwise, the topology on the set would be assumed the

usual topology

induced by norms on

ℝ

n

.

•

A

random variable

X

is discrete if and only if its image space is a discrete set (which by what’s just been said means that the image is a discrete topological space for some topology specified by the context). The most common example by far is a random variable taking its values in a enumerated set (e.g. the values of a die, or a set of possible answers to a question in a survey).

•

Discretization of ODEs and PDEs is the process of converting

equations

on functions on

open sets

of

ℝ

n

(with

boundary conditions

) into equations on functions on discrete subsets of

ℝ

n

.

Title

discrete

Canonical name

Discrete

Date of creation

2013-03-22 17:56:49

Last modified on

2013-03-22 17:56:49

Owner

lalberti (18937)

Last modified by

lalberti (18937)

Numerical id

8

Author

lalberti (18937)

Entry type

Definition

Classification

msc 54A05

Related topic

Discrete

Generated on Sat Feb 10 11:12:46 2018 by

LaTeXML
finite complement topology

finite complement topology

Let

X

be a set. We can define the

finite

complement

topology

on

X

by declaring a subset

U

⊂

X

to be open if

X

\

U

is finite, or if

U

is all of

X

or the

empty set

. Note that this is

equivalent

to defining a topology by defining the

closed sets

in

X

to be all

finite sets

(and

X

itself).

If

X

is finite, the finite complement topology on

X

is clearly the

discrete topology

, as the complement of

any

subset is finite.

If

X

is

countably infinite

(or larger), the finite complement topology gives a standard example of a space that is not

Hausdorff

(each

open set

must contain all but finitely many points, so any two open sets must

intersect

).

In general, the finite complement topology on an

infinite set

satisfies

strong

compactness

conditions (

compact

,

σ

-compact (

http://planetmath.org/SigmaCompact

),

sequentially compact

, etc.) since each open set in a cover contains ”almost all” of the points of

X

. On the other hand, the finite complement topology fails all but the simplest of

separation axioms

since, as above,

X

is

hyperconnected

under this topology.

The finite complement topology is the coarsest T1-topology on a given set.

Title

finite complement topology

Canonical name

FiniteComplementTopology

Date of creation

2013-03-22 14:37:54

Last modified on

2013-03-22 14:37:54

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

8

Author

mathcam (2727)

Entry type

Definition

Classification

msc 54A05

Synonym

cofinite topology

Related topic

CofiniteTopology

Related topic

CofiniteAndCocountableTopology

Generated on Sat Feb 10 11:12:47 2018 by

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F_σ set

F

σ

set

A subset of a

topological space

is called a

F

σ

set if it equals the union of a

countable

collection

of

closed sets

.

The

complement

of a

F

σ

set is a

G

δ

set (

http://planetmath.org/G_DeltaSet

).

For

instance

, the

X

set of all points

(

x

,

y

)

in the plane such that either

y

=

0

or

x

/

y

is

rational

is an

F

σ

set because it can be expressed as the union of a

countable set

of lines:

X

=

{

(

x

,

0

)

∣

x

∈

ℝ

}

∪

⋃

r

∈

ℚ

{

(

r

⁢

y

,

y

)

∣

y

∈

ℝ

}

Title

F

σ

set

Canonical

name

FsigmaSet

Date of creation

2013-03-22 14:37:59

Last modified on

2013-03-22 14:37:59

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

9

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 54A05

Related topic

G_DeltaSet

Related topic

G_deltaSet

Related topic

PavedSet

Related topic

PavedSpace

Generated on Sat Feb 10 11:12:49 2018 by

LaTeXML
G_δ set

G

δ

set

A

G

δ

set is a set which can be expressed as the

intersection

of a

countable

collection

of

open sets

.

The

complement

of a

G

δ

set is an

F

σ

set (

http://planetmath.org/F_sigmaSet

).

For example, the

closed interval

[

-

1

,

+

1

]

∈

ℝ

is a

G

δ

set because

[

-

1

,

+

1

]

=

⋂

n

=

1

∞

(

-

1

n

-

1

,

1

n

+

1

)

Title

G

δ

set

Canonical name

GdeltaSet

Date of creation

2013-03-22 14:38:02

Last modified on

2013-03-22 14:38:02

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

7

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 54A05

Related topic

F_sigmaSet

Related topic

PavedSet

Related topic

PavedSpace

Generated on Sat Feb 10 11:12:51 2018 by

LaTeXML
generic point

generic point

Let

X

be a

topological space

and suppose that

x

∈

X

. If for all

y

∈

X

\

{

x

}

we have

x

∉

{

y

}

¯

then we say that

x

is a

generic point

. Equivalently,

x

is

generic

iff it is not a

specialization

point of any point in

X

, other than

x

itself.

Title

generic point

Canonical name

GenericPoint

Date of creation

2013-03-22 16:22:28

Last modified on

2013-03-22 16:22:28

Owner

jocaps (12118)

Last modified by

jocaps (12118)

Numerical id

8

Author

jocaps (12118)

Entry type

Definition

Classification

msc 54A05

Defines

generic point

Defines

generic

Generated on Sat Feb 10 11:12:53 2018 by

LaTeXML