aarjaneiro/planetmath huggingface dataset:
boundary of an open set is nowhere dense

boundary of an open set is nowhere dense

This entry provides another example of a nowhere

dense set

.

Proposition 1

.

If

A

is an

open set

in a

topological space

X

, then

∂

⁡

A

, the

boundary

of

A

is nowhere dense.

Proof.

Let

B

=

∂

⁡

A

. Since

B

=

A

¯

∩

A

∁

¯

, it is closed, so all we need to show is that

B

has empty

interior

int

⁡

(

B

)

=

∅

. First notice that

B

=

A

¯

∩

A

∁

, since

A

is open. Now, we invoke one of the

interior axioms

, namely

int

⁡

(

U

∩

V

)

=

int

⁡

(

U

)

∩

int

⁡

(

V

)

. So, by direct

computation

, we have

int

⁡

(

B

)

=

int

⁡

(

A

¯

)

∩

int

⁡

(

A

∁

)

=

int

⁡

(

A

¯

)

∩

A

¯

∁

⊆

A

¯

∩

A

¯

∁

=

∅

.

The second equality and the

inclusion

follow from the general

properties

of the interior

operation

, the proofs of which can be found here (

http://planetmath.org/DerivationOfPropertiesOnInteriorOperation

).
∎

Remark

. The fact that

A

is open is essential. Otherwise, the

proposition

fails in general. For example, the

rationals

ℚ

, as a subset of the reals

ℝ

under the usual

order topology

, is not open, and its boundary is not nowhere dense, as

ℚ

¯

∩

ℚ

∁

¯

=

ℝ

∩

ℝ

=

ℝ

, whose interior is

ℝ

itself, and thus not empty.

Title

boundary of an open set is nowhere dense

Canonical

name

BoundaryOfAnOpenSetIsNowhereDense

Date of creation

2013-03-22 17:55:41

Last modified on

2013-03-22 17:55:41

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Derivation

Classification

msc 54A99

Generated on Sat Feb 10 11:15:04 2018 by

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box topology

box topology

Let

{

(

X

α

,

𝒯

α

)

}

α

∈

A

be a family of

topological spaces

.
Let

Y

denote the

generalized Cartesian product

of the sets

X

α

,
that is

Y

=

∏

α

∈

A

X

α

.

Let

ℬ

denote the set of all

products

of

open sets

of the corresponding
spaces, that is

ℬ

=

{

∏

α

∈

A

U

α

|

U

α

∈

𝒯

α

⁢

for all

⁢

α

∈

A

}

.

Now we can construct the

box product

(

Y

,

𝒮

)

, where

𝒮

,
referred to as the

box topology

,
is the

topology

the base

ℬ

.

When

A

is a finite (

http://planetmath.org/Finite

) set,
the box topology coincides with the product topology.

Example

As an example,
the box product of two topological spaces

(

X

0

,

𝒯

0

)

and

(

X

1

,

𝒯

1

)

is

(

X

0

×

X

1

,

𝒮

)

,
where the box topology

𝒮

(which is the same as the product topology)
consists of all sets of the form

⋃

i

∈

I

(

U

i

×

V

i

)

,
where

I

is some

index set

and for each

i

∈

I

we have

U

i

∈

𝒯

0

and

V

i

∈

𝒯

1

.

Title

box topology

Canonical

name

BoxTopology

Date of creation

2013-03-22 12:46:55

Last modified on

2013-03-22 12:46:55

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

9

Author

yark (2760)

Entry type

Definition

Classification

msc 54A99

Synonym

box product topology

Related topic

ProductTopology

Defines

box product

Generated on Sat Feb 10 11:15:06 2018 by

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closure

closure

The

closure

A

¯

of a subset

A

of a

topological space

X

is the

intersection

of all

closed sets

containing

A

.

Equivalently,

A

¯

consists of

A

together with all

limit points

of

A

in

X

or equivalently

x

∈

A

¯

if and only if every

neighborhood

of

x

intersects

A

. Sometimes the notation

cl

⁡

(

A

)

is used.

If it is not clear, which topological space is used, one writes

A

¯

X

. Note that if

Y

is a

subspace

of

X

, then

A

¯

X

may not be the same as

A

¯

Y

. For example, if

X

=

ℝ

,

Y

=

(

0

,

1

)

and

A

=

(

0

,

1

)

, then

A

¯

X

=

[

0

,

1

]

while

A

¯

Y

=

(

0

,

1

)

.

Title

closure

Canonical

name

Closure

Date of creation

2013-03-22 12:05:40

Last modified on

2013-03-22 12:05:40

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

9

Author

mathwizard (128)

Entry type

Definition

Classification

msc 54A99

Related topic

ClosureAxioms

Related topic

Interior

Generated on Sat Feb 10 11:15:08 2018 by

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comparison of filters

comparison of filters

Let

𝔽

1

and

𝔽

2

be two filters on the same set. The following terminology is commonly used to describe the relation of

𝔽

1

to

𝔽

2

:

𝔽

2

is said to be

finer

than

𝔽

1

if

𝔽

1

⊆

𝔽

2

.

𝔽

2

is said to be

coarser

than

𝔽

1

if

𝔽

1

⊇

𝔽

2

.

𝔽

2

is said to be

strictly finer

than

𝔽

1

if

𝔽

1

⊂

𝔽

2

.

𝔽

2

is said to be

strictly coarser

than

𝔽

1

if

𝔽

1

⊃

𝔽

2

.

𝔽

1

and

𝔽

2

are said to be

comparable

if either

𝔽

1

⊆

𝔽

2

or

𝔽

1

⊇

𝔽

2

.

Title

comparison of filters

Canonical

name

ComparisonOfFilters

Date of creation

2013-03-22 14:41:38

Last modified on

2013-03-22 14:41:38

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

5

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 54A99

Classification

msc 03E99

Defines

finer

Defines

coarser

Defines

strictly finer

Defines

strictly coarser

Defines

comparable

Generated on Sat Feb 10 11:15:10 2018 by

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cover

cover

Definition

(

[

1

]

, pp. 49)
Let

Y

be a subset of a set

X

. A

cover

for

Y

is a

collection

of sets

𝒰

=

{

U

i

}

i

∈

I

such that each

U

i

is a subset of

X

, and

Y

⊂

⋃

i

∈

I

U

i

.

The collection of sets can be arbitrary, that is,

I

can be
finite,

countable

, or

uncountable

. The cover is correspondingly called a

finite cover

,

countable cover

, or

uncountable cover

.

A

subcover

of

𝒰

is a subset

𝒰

′

⊂

𝒰

such that

𝒰

′

is also a cover of

X

.

A

refinement

𝒱

of

𝒰

is a cover of

X

such that for every

V

∈

𝒱

there is some

U

∈

𝒰

such that

V

⊂

U

. When

𝒱

refines

𝒰

, it is usually written

𝒱

⪯

𝒰

.

⪯

is a preorder on the set of covers of any

topological space

X

.

If

X

is a topological space and the members of

𝒰

are

open sets

,
then

𝒰

is said to be an

open cover

.
Open subcovers and open refinements are defined similarly.

Examples

1.

If

X

is a set, then

{

X

}

is a cover of

X

.

2.

The

power set

of a set

X

is a cover of

X

.

3.

A

topology

for a set is a cover of that set.

References

1

J.L. Kelley,

General Topology

,
D. van Nostrand Company, Inc., 1955.

Title

cover

Canonical name

Cover

Date of creation

2013-03-22 12:06:31

Last modified on

2013-03-22 12:06:31

Owner

mps (409)

Last modified by

mps (409)

Numerical id

19

Author

mps (409)

Entry type

Definition

Classification

msc 54A99

Related topic

Compact

Related topic

VarepsilonNet

Related topic

Site

Related topic

CoveringSpace

Related topic

CompactMetricSpacesAreSecondCountable

Defines

open cover

Defines

subcover

Defines

refinement

Defines

finite cover

Defines

countable cover

Defines

uncountable cover

Defines

open subcover

Defines

open refinement

Defines

cover refinement

Generated on Sat Feb 10 11:15:27 2018 by

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dense in-itself

dense in-itself

A subset

A

of a

topological space

is said to be

dense-in-itself

if

A

contains no

isolated points

.

Note that if the subset

A

is also a

closed set

, then

A

will be a

perfect set

.

Conversely

, every perfect set is dense-in-itself.

A simple example of a set which is dense-in-itself but not closed (and hence not a perfect set) is the subset of

irrational numbers

. This set is dense-in-itself because every

neighborhood

of an irrational number

x

contains at least one other irrational number

y

≠

x

. On the other hand, this set of irrationals is not closed because every

rational number

lies in its

closure

.

For

similar

reasons, the set of rational numbers is also dense-in-itself but not closed.

Title

dense in-itself

Canonical name

DenseInitself

Date of creation

2013-03-22 14:38:29

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2013-03-22 14:38:29

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

4

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 54A99

Related topic

ScatteredSpace

Generated on Sat Feb 10 11:15:29 2018 by

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dense set

dense set

A subset

D

of a

topological space

X

is said to be

dense

(or

everywhere dense

) in

X

if the

closure

of

D

is equal to

X

.
Equivalently,

D

is dense if and only if

D

intersects

every nonempty

open set

.

In the special case that

X

is a

metric space

with metric

d

,
then this can be rephrased as:
for all

ε

>

0

and all

x

∈

X

there is

y

∈

D

such that

d

⁢

(

x

,

y

)

<

ε

.

For example, both the

rationals

ℚ

and the irrationals

ℝ

∖

ℚ

are dense in the reals

ℝ

.

The least

cardinality

of a

dense set

of a topological space
is called the

density

of the space.
It is conventional to take the density to be

ℵ

0

if it would otherwise be finite;
with this convention,
the spaces of density

ℵ

0

are precisely the

separable spaces

.
The density of a topological space

X

is denoted

d

⁢

(

X

)

.
If

X

is a

Hausdorff space

,
it can be shown that

|

X

|

≤

2

2

d

⁢

(

X

)

.

Title

dense set

Canonical name

DenseSet

Date of creation

2013-03-22 12:05:42

Last modified on

2013-03-22 12:05:42

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

12

Author

yark (2760)

Entry type

Definition

Classification

msc 54A99

Synonym

dense subset

Synonym

everywhere dense set

Synonym

everywhere dense subset

Synonym

everywhere-dense set

Synonym

everywhere-dense subset

Related topic

NowhereDense

Related topic

DenseInAPoset

Defines

dense

Defines

everywhere dense

Defines

everywhere-dense

Defines

density

Generated on Sat Feb 10 11:15:32 2018 by

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examples of filters

examples of filters

•

If

X

is any set and

A

⊆

X

then

ℱ

=

{

F

⊆

X

:

A

⊆

F

}

is a

fixed filter

on

X

;

ℱ

is an

ultrafilter

iff

A

consists of a single point.

•

If

X

is any

infinite set

, then

{

F

⊆

X

:

X

∖

F

⁢

is finite

}

is a free filter on

X

, called the

cofinite filter

.

•

The filter on

ℝ

generated by the

filter base

{

(

n

,

∞

)

:

n

∈

ℕ

}

is called the

Fréchet filter

on

ℝ

; it is a free filter which does not

converge

or have any

accumulation points

.

•

The filter on

ℝ

generated by the filter base

{

(

0

,

ε

)

:

ε

>

0

}

is a free filter on

ℝ

which converges to

0

.

Title

examples of filters

Canonical name

ExamplesOfFilters

Date of creation

2013-03-22 12:54:31

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2013-03-22 12:54:31

Owner

Evandar (27)

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Evandar (27)

Numerical id

8

Author

Evandar (27)

Entry type

Example

Classification

msc 54A99

Classification

msc 03E99

Defines

cofinite filter

Defines

Fréchet filter

Generated on Sat Feb 10 11:15:33 2018 by

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examples of nowhere dense sets

examples of nowhere dense sets

Note that

ℤ

is nowhere dense in

ℝ

under the

usual topology

:

int

⁡

ℤ

¯

=

int

⁡

ℤ

=

∅

. Similarly,

1

n

⁢

ℤ

is nowhere dense for every

n

∈

ℤ

with

n

>

0

.

This result provides an alternative way to prove that

ℚ

is meager in

ℝ

under the usual topology, since

ℚ

=

⋃

n

∈

ℤ

⁢

and

⁢

n

>

0

1

n

⁢

ℤ

.

Title

examples of nowhere dense sets

Canonical name

ExamplesOfNowhereDenseSets

Date of creation

2013-03-22 17:07:05

Last modified on

2013-03-22 17:07:05

Owner

Wkbj79 (1863)

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Numerical id

4

Author

Wkbj79 (1863)

Entry type

Example

Classification

msc 54A99

Related topic

ExampleOfAMeagerSet

Generated on Sat Feb 10 11:15:36 2018 by

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identification topology

identification topology

Let

f

be a

function

from a

topological space

X

to a set

Y

. The

identification topology

on

Y

with respect to

f

is defined to be the finest

topology

on

Y

such that the function

f

is

continuous

.

Theorem 1

.

Let

f

:

X

→

Y

be defined as above. The following are

equivalent

:

1.

𝒯

is the identification topology on

Y

.

2.

U

⊆

Y

is open under

𝒯

iff

f

-

1

⁢

(

U

)

is open in

X

.

Proof.

(

1

.

⇒

2

.

) If

U

is open under

𝒯

, then

f

-

1

⁢

(

U

)

is open in

X

as

f

is continuous under

𝒯

. Now, suppose

U

is not open under

𝒯

and

f

-

1

⁢

(

U

)

is open in

X

. Let

ℬ

be a

subbase

of

𝒯

. Define

ℬ

′

:=

ℬ

∪

{

U

}

. Then the topology

𝒯

′

generated by

ℬ

′

is a

strictly finer

topology than

𝒯

making

f

continuous, a

contradiction

.

(

2

.

⇒

1

.

) Let

𝒯

be the topology defined by 2. Then

f

is continuous. Suppose

𝒯

′

is another topology on

Y

making

f

continuous. Let

U

be

𝒯

′

-open. Then

f

-

1

⁢

(

U

)

is open in

X

, which implies

U

is

𝒯

-open. Thus

𝒯

′

⊆

𝒯

and

𝒯

is finer than

𝒯

′

.
∎

Remarks

.

•

𝒮

=

{

f

⁢

(

V

)

∣

V

⁢

is open in

⁢

X

}

is a subbasis for

f

⁢

(

X

)

, using the

subspace topology

on

f

⁢

(

X

)

of the identification topology on

Y

.

•

More generally, let

X

i

be a family of topological spaces and

f

i

:

X

i

→

Y

be a family of functions from

X

i

into

Y

. The

identification topology

on

Y

with respect to the family

f

i

is the finest topology on

Y

making each

f

i

a

continuous function

. In literature, this topology is also called the

final topology

.

•

The dual

concept

of this is the

initial topology

.

•

Let

f

:

X

→

Y

be defined as above. Define

binary relation

∼

on

X

so that

x

∼

y

iff

f

⁢

(

x

)

=

f

⁢

(

y

)

. Clearly

∼

is an

equivalence relation

. Let

X

*

be the quotient

X

/

∼

. Then

f

induces an injective map

f

*

:

X

*

→

Y

given by

f

*

⁢

(

[

x

]

)

=

f

⁢

(

x

)

. Let

Y

be given the identification topology and

X

*

the

quotient topology

(induced by

∼

), then

f

*

is continuous. Indeed, for if

V

⊆

Y

is open, then

f

-

1

⁢

(

V

)

is open in

X

. But then

f

-

1

⁢

(

V

)

=

⋃

f

*

-

1

⁢

(

V

)

, which implies

f

*

-

1

⁢

(

V

)

is open in

X

*

. Furthermore, the

argument

is reversible, so that if

U

is open in

X

*

, then so is

f

*

⁢

(

U

)

open in

Y

. Finally, if

f

is

surjective

, so is

f

*

, so that

f

*

is a

homeomorphism

.

Title

identification topology

Canonical name

IdentificationTopology

Date of creation

2013-03-22 14:41:26

Last modified on

2013-03-22 14:41:26

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

11

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 54A99

Synonym

final topology

Related topic

InitialTopology

Generated on Sat Feb 10 11:15:40 2018 by

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limit point

limit point

Let

X

be a

topological space

, and let

A

⊆

X

. An element

x

∈

X

is said to be a

limit point

of

A

if every

open set

containing

x

also contains at least one point of

A

distinct from

x

. Note that we can often take a nested sequence of open such sets, and can thereby construct a sequence of points which converge to

x

, partially motivating the terminology ”limit” in this case.

Equivalently:

•

x

is a limit point of

A

if and only if there is a net in

A

converging to

x

which is not

residually constant

.

•

x

is a limit point of

A

if and only if there is a filter on

A

converging (

http://planetmath.org/filter

) to

x

.

•

If

X

is a metric (or

first countable

) space,

x

is a limit point of

A

if and only if there is a sequence of points in

A

∖

{

x

}

converging to

x

.

Title

limit point

Canonical name

LimitPoint

Date of creation

2013-03-22 12:06:51

Last modified on

2013-03-22 12:06:51

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

15

Author

mathcam (2727)

Entry type

Definition

Classification

msc 54A99

Synonym

accumulation point

Synonym

cluster point

Related topic

AlternateStatementOfBolzanoWeierstrassTheorem

Generated on Sat Feb 10 11:15:55 2018 by

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limit points of uncountable subsets of R^n

limit points of uncountable subsets of R^n

Proposition

.

Let

ℝ

n

be an

n

-dimensional, real

normed space

and let

A

⊆

ℝ

n

. If

A

is

uncountable

, then there exists

limit point

of

A

in

ℝ

n

.

Proof.

For any

k

∈

ℕ

let

𝔹

k

=

{

v

∈

ℝ

n

|

||

v

||

≤

k

}

,

i.e.

𝔹

k

is a

closed ball

centered in

0

with radius

k

. Assume, that for any

k

the set

V

k

=

𝔹

k

∩

A

is finite. Then

⋃

V

k

=

A

would be at most

countable

.

Contradiction

, since

A

is uncountable. Thus, there exists

k

0

∈

ℕ

such that

V

k

0

is

infinite

. But

V

k

0

⊆

𝔹

k

0

and since

𝔹

k

0

is

compact

(and

V

k

0

is infinite), then there exists limit point of

V

k

0

in

ℝ

n

. This

completes

the proof.

□

Corollary.

If

A

⊆

ℝ

n

is uncountable, then there exist infinitely many limit points of

A

in

ℝ

n

.

Proof.

Assume, that there are finitely many limit points of

A

, namely

x

1

,

…

,

x

k

∈

ℝ

n

. For

ε

>

0

define

A

ε

=

{

v

∈

ℝ

n

|

∀

i

||

v

-

x

i

||

>

ε

}

.

Briefly speaking,

A

ε

is a

complement

of a union of closed balls centered at

x

i

with radii

ε

. Of course

A

ε

≠

∅

since there are finitely many limit points. Let

V

ε

=

A

∩

A

ε

.

Assume, that

V

ε

is countable for every

ε

. Then

A

⊆

⋃

n

∈

ℕ

V

1

n

∪

{

x

1

,

…

,

x

k

}

would be at most countable (of course under

assumption

of

Axiom of Choice

). Contradiction. Thus, there is

γ

>

0

such that

V

γ

is uncountable. Then (due to proposition) there is a limit point

x

′

∈

ℝ

n

of

V

γ

. Note, that

x

′

∈

V

γ

¯

⊆

V

γ

′

for some

0

<

γ

′

<

γ

. Thus

x

′

is different from any

x

i

. Contradiction, since

x

′

is also a limit point of

A

.

□

Title

limit points of uncountable subsets of R^n

Canonical name

LimitPointsOfUncountableSubsetsOfRn

Date of creation

2013-03-22 19:07:57

Last modified on

2013-03-22 19:07:57

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

6

Author

joking (16130)

Entry type

Theorem

Classification

msc 54A99

Generated on Sat Feb 10 11:15:58 2018 by

LaTeXML
neighborhood retract

neighborhood retract

Let

X

be a

topological space

and

Y

a

subspace

of

X

. If there exists an

open set

U

such that

Y

⊂

U

⊂

X

and

Y

is a

retract

of

U

, then we say

Y

is a

neighborhood retract

of

X

.

Title

neighborhood retract

Canonical name

NeighborhoodRetract

Date of creation

2013-03-22 14:39:41

Last modified on

2013-03-22 14:39:41

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

5

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 54A99

Generated on Sat Feb 10 11:15:59 2018 by

LaTeXML
nonempty perfect subset of ℝ that contains no rational number, a

nonempty perfect subset of

ℝ

that contains no rational number, a

We will construct a nonempty

perfect set

contained

in

ℝ

that
contains no

rational number

.

We will begin with a

closed interval

, and then, imitating the
construction of Cantor set, we will inductively delete each
rational number in it together with an open interval. We will do
it in such a way that the

end points

of the open intervals will
never be deleted afterwards.

Let

E

0

=

[

b

0

,

a

0

]

for some irrational numbers

a

0

and

b

0

,
with

b

0

<

a

0

. Let

{

q

1

,

q

2

,

q

3

,

…

}

be an enumeration of the
rational numbers in

[

b

0

,

a

0

]

. For each

q

i

, we will define an
open interval

(

a

i

,

b

i

)

and delete it.

Let

a

1

and

b

1

be two irrational numbers such that

b

0

<

a

1

<

q

1

<

b

1

<

a

0

. Define

E

1

=

E

0

\

(

a

1

,

b

1

)

.

Having defined

E

1

,

E

2

,

…

,

E

n

,

a

1

,

a

2

,

…

,

a

n

and

b

1

,

b

2

,

…

,

b

n

, let’s define

a

n

+

1

and

b

n

+

1

:

If

q

n

+

1

∈

⋃

k

=

1

n

(

a

k

,

b

k

)

then there exists
an

i

≤

n

such that

q

n

+

1

∈

(

a

i

,

b

i

)

. Let

a

n

+

1

=

a

i

and

b

n

+

1

=

b

i

.

Otherwise let

a

n

+

1

and

b

n

+

1

be two irrational numbers
such that

b

0

<

a

n

+

1

<

q

n

+

1

<

b

n

+

1

<

a

0

, and which

satisfy

:

q

n

+

1

-

a

n

+

1

<

min

i

=

0

,

1

,

2

,

…

,

n

⁡

{

|

q

n

+

1

-

b

i

|

}

and

b

n

+

1

-

q

n

+

1

<

min

i

=

0

,

1

,

2

,

…

,

n

⁡

{

|

a

i

-

q

n

+

1

|

}

.

Now define

E

n

+

1

=

E

n

\

(

a

n

+

1

,

b

n

+

1

)

. Note that
by our choice of

a

n

+

1

and

b

n

+

1

any of the previous end
points are not removed from

E

n

.

Let

E

=

⋂

n

=

1

∞

E

n

.

E

is clearly
nonempty, does not contain any rational number, and also it is

compact

, being an

intersection

of compact sets.

Now let us see that

E

does not have any

isolated points

. Let

x

∈

E

, and

ϵ

>

0

be given. If

x

≠

a

j

for any

j

∈

{

0

,

1

,

2

,

…

}

, choose a rational number

q

k

such that

x

<

q

k

<

x

+

ϵ

. Then

q

k

∈

(

a

k

,

b

k

)

and
since

x

∈

E

we must have

x

<

a

k

, which means

a

k

∈

(

x

,

x

+

ϵ

)

. Since we know that

a

k

∈

E

, this shows that

x

is a

limit point

. Otherwise, if

x

=

a

j

for some

j

, then choose a

q

k

such that

x

-

ϵ

<

q

k

<

x

. Similarly,

q

k

∈

(

a

k

,

b

k

)

and it follows that

b

k

∈

(

x

-

ϵ

,

x

)

. We have shown that
any point of

E

is a limit point, hence

E

is

perfect

.

Title

nonempty perfect subset of

ℝ

that contains no rational number, a

Canonical name

NonemptyPerfectSubsetOfmathbbRThatContainsNoRationalNumberA

Date of creation

2013-03-22 15:26:10

Last modified on

2013-03-22 15:26:10

Owner

Gorkem (3644)

Last modified by

Gorkem (3644)

Numerical id

18

Author

Gorkem (3644)

Entry type

Example

Classification

msc 54A99

Generated on Sat Feb 10 11:16:05 2018 by

LaTeXML
nowhere dense

nowhere dense

In a

topological space

X

, a set

A

is called

nowhere dense

if the

interior

of its

closure

is empty:

int

⁡

A

¯

=

∅

.

Title

nowhere dense

Canonical name

NowhereDense

Date of creation

2013-03-22 12:43:35

Last modified on

2013-03-22 12:43:35

Owner

ariels (338)

Last modified by

ariels (338)

Numerical id

5

Author

ariels (338)

Entry type

Definition

Classification

msc 54A99

Generated on Sat Feb 10 11:16:07 2018 by

LaTeXML
perfect set

perfect set

A set is called

perfect

if it is equal to the set of its

limit points

. An non-trivial example of a perfect set is the

http://planetmath.org/node/2083

middle-thirds Cantor set. In fact a more general class of sets is referred to as Cantor sets, which all have (among others) the property of being perfect.

Title

perfect set

Canonical name

PerfectSet

Date of creation

2013-03-22 13:18:51

Last modified on

2013-03-22 13:18:51

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

7

Author

mathwizard (128)

Entry type

Definition

Classification

msc 54A99

Defines

perfect

Generated on Sat Feb 10 11:16:09 2018 by

LaTeXML
point-finite

point-finite

A

collection

𝒰

of subsets of a

topological space

X

is said to be

point-finite

if every point of

X

lies in only finitely many members of

𝒰

.

Compare this to the

stronger

property of being

locally finite

.

Point-finiteness is used in the definition of

metacompactness

.

Title

point-finite

Canonical name

Pointfinite

Date of creation

2013-03-22 16:17:02

Last modified on

2013-03-22 16:17:02

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

5

Author

yark (2760)

Entry type

Definition

Classification

msc 54A99

Synonym

point finite

Related topic

LocallyFinite

Related topic

Metacompact

Defines

point finite collection

Defines

point-finite collection

Defines

point finiteness

Defines

point-finiteness

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

LaTeXML