aarjaneiro/planetmath huggingface dataset:
boundary / frontier

boundary / frontier

Definition.

Let

X

be a

topological space

and let

A

be a subset
of

X

. The

boundary

(or

frontier

) of

A

is the set

∂

⁡

A

=

A

¯

∩

X

\

A

¯

,
where the overline denotes the

closure

of a set.
Instead of

∂

⁡

A

, many authors use some other notation
such as

bd

⁡

(

A

)

,

fr

⁡

(

A

)

,

A

b

or

β

⁢

(

A

)

.
Note that the

∂

symbol is also used for other meanings of ‘boundary’.

From the definition, it follows that the boundary of any set is a

closed set

.
It also follows that

∂

⁡

A

=

∂

⁡

(

X

\

A

)

,
and

∂

⁡

X

=

∅

=

∂

⁡

∅

.

The term ‘boundary’ (but not ‘frontier’) is used in a different sense for

topological manifolds

: the boundary

∂

⁡

M

of a topological

n

-

manifold

M

is the set of points in

M

that do not have a

neighbourhood

homeomorphic

to

ℝ

n

. (Some authors define topological manifolds in such a way that they necessarily have empty boundary.)
For example, the boundary of the topological

1

-manifold

[

0

,

1

]

is

{

0

,

1

}

.

Title

boundary / frontier

Canonical name

BoundaryFrontier

Date of creation

2013-03-22 13:34:46

Last modified on

2013-03-22 13:34:46

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

17

Author

yark (2760)

Entry type

Definition

Classification

msc 54-00

Synonym

boundary

Synonym

frontier

Synonym

topological boundary

Related topic

ExtendedBoundary

Related topic

Interior

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

LaTeXML
closed set

closed set

\PMlinkescapephrase

closed under

Let

(

X

,

τ

)

be a

topological space

. Then a subset

C

⊆

X

is

closed

if its complement

X

∖

C

is open under the

topology

τ

.

Examples:

•

In any topological space

(

X

,

τ

)

, the sets

X

and

∅

are always closed.

•

Consider

ℝ

with the

standard topology

. Then

[

0

,

1

]

is closed since its complement

(

-

∞

,

0

)

∪

(

1

,

∞

)

is open (being the union of two

open sets

).

•

Consider

ℝ

with the

lower limit topology

. Then

[

0

,

1

)

is closed since its complement

(

-

∞

,

0

)

∪

[

1

,

∞

)

is open.

Closed subsets

can also be characterized as follows:

A subset

C

⊆

X

is closed if and only if

C

contains all of its

cluster points

, that is,

C

′

⊆

C

.

So the set

{

1

,

1

/

2

,

1

/

3

,

1

/

4

,

…

}

is not closed under the standard topology on

ℝ

since

0

is a cluster point not contained in the set.

Title

closed set

Canonical

name

ClosedSet

Date of creation

2013-03-22 12:30:23

Last modified on

2013-03-22 12:30:23

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

10

Author

yark (2760)

Entry type

Definition

Classification

msc 54-00

Synonym

closed subset

Defines

closed

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

LaTeXML
coarser

coarser

The set of

topologies

which can be defined on a set is partially ordered under

inclusion

. Below, we list several synonymous terms which are used to refer to this order. Let

𝒰

and

𝒱

be two topologies defined on a set

E

. All of the following

expressions

mean that

𝒰

⊂

𝒱

:

•

𝒰

is

weaker

than

𝒱

•

𝒰

is

coarser

than

𝒱

•

𝒱

is

finer

than

𝒰

•

𝒱

is a

refinement

of

𝒰

•

𝒱

is an

expansion

of

𝒰

It is worth noting that this condition is

equivalent

to the requirement that the

identity map

from

(

E

,

𝒱

)

to

(

E

,

𝒰

)

is

continuous

.

Title

coarser

Canonical

name

Coarser

Date of creation

2013-03-22 12:56:03

Last modified on

2013-03-22 12:56:03

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

10

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 54-00

Synonym

stronger

Related topic

InitialTopology

Related topic

LatticeOfTopologies

Defines

weaker

Defines

finer

Defines

refinement

Defines

expansion

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

LaTeXML
compact-open topology

compact-open topology

Let

X

and

Y

be

topological spaces

, and let

C

⁢

(

X

,

Y

)

be the set of

continuous maps

from

X

to

Y

.

Given a

compact

subspace

K

of

X

and an

open set

U

in

Y

,

let

𝒰

K

,

U

:=

{

f

∈

C

⁢

(

X

,

Y

)

:

f

⁢

(

x

)

∈

U

⁢

whenever

⁢

x

∈

K

}

.

Define the

compact-open topology

on

C

⁢

(

X

,

Y

)

to be the

topology

generated by the

subbasis

{

𝒰

K

,

U

:

K

⊂

X

⁢

compact,

U

⊂

Y

⁢

open

}

.

If

Y

is a

uniform space

(for example, if

Y

is a

metric space

), then this is the topology of uniform convergence on compact sets. That is, a sequence

(

f

n

)

converges

to

f

in the compact-open topology if and only if for every compact subspace

K

of

X

,

(

f

n

)

converges to

f

uniformly on

K

. If in addition

X

is a

compact space

, then this is the topology of uniform convergence.

Title

compact-open topology

Canonical name

CompactopenTopology

Date of creation

2013-03-22 13:25:26

Last modified on

2013-03-22 13:25:26

Owner

antonio (1116)

Last modified by

antonio (1116)

Numerical id

8

Author

antonio (1116)

Entry type

Definition

Classification

msc 54-00

Synonym

topology of compact convergence

Related topic

UniformConvergence

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

LaTeXML
completely normal

completely normal

Let

X

be a

topological space

.

X

is said to be

if whenever

A

,

B

⊆

X

with

A

∩

B

¯

=

A

¯

∩

B

=

∅

, then there are disjoint

open sets

U

and

V

such that

A

⊆

U

and

B

⊆

V

.

Equivalently, a topological space

X

is if and only if every

subspace

is normal.

Title

completely normal

Canonical name

CompletelyNormal

Date of creation

2013-03-22 12:13:51

Last modified on

2013-03-22 12:13:51

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

7

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 54-00

Synonym

complete normality

Related topic

NormalTopologicalSpace

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

LaTeXML
completely separated

completely separated

Proposition 1

.

Let

A

,

B

be two subsets of a

topological space

X

. The following are

equivalent

:

1.

There is a

continuous function

f

:

X

→

[

0

,

1

]

such that

f

⁢

(

A

)

=

0

and

f

⁢

(

B

)

=

1

,

2.

There is a continuous function

g

:

X

→

ℝ

such that

g

⁢

(

A

)

≤

r

<

s

≤

g

⁢

(

B

)

,

r

,

s

∈

ℝ

.

Proof.

Clearly 1 implies 2 (by setting

r

=

0

and

s

=

1

). To see that 2 implies 1, first take the

transformation

h

⁢

(

x

)

=

g

⁢

(

x

)

-

r

s

-

r

so that

h

⁢

(

A

)

≤

0

<

1

≤

h

⁢

(

B

)

. Then take the transformation

f

⁢

(

x

)

=

(

h

⁢

(

x

)

∨

0

)

∧

1

, where

0

⁢

(

x

)

=

0

and

1

⁢

(

x

)

=

1

for all

x

∈

X

. Then

f

⁢

(

A

)

=

(

h

⁢

(

A

)

∨

0

)

∧

1

=

0

∧

1

=

0

and

f

⁢

(

B

)

=

(

h

⁢

(

B

)

∨

0

)

∧

1

=

h

⁢

(

B

)

∧

1

=

1

. Here,

∨

and

∧

denote the

binary operations

of taking the

maximum

and

minimum

of two given

real numbers

(see

ring of continuous functions

for more detail). Since during each transformation, the resulting

function

remains

continuous

, the first assertion is proved.
∎

Definition

. Any two sets

A

,

B

in a topological space

X

satisfying the above equivalent conditions are said to be

completely separated

. When

A

and

B

are completely separated, we also say that

{

A

,

B

}

is completely separated.

Clearly, two sets that are completely separated are

disjoint

, and in fact

separated

.

Remark

. A T1 topological space in which every pair of disjoint

closed sets

are completely separated is a

normal space

. A T0 topological space in which every pair consisting of a closed set and a

singleton

is completely separated is a

completely regular space

.

Title

completely separated

Canonical

name

CompletelySeparated

Date of creation

2013-03-22 16:54:51

Last modified on

2013-03-22 16:54:51

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 54-00

Classification

msc 54D05

Classification

msc 54D15

Synonym

functionally distinguishable

Generated on Sat Feb 10 11:10:31 2018 by

LaTeXML
continuous proper map

continuous proper map

Let

X

,

Y

be

topological spaces

and

p

:

X

→

Y

a

continuous map

.
The map

p

is called

continuous proper

if for every

compact

K

⊆

Y

the

preimage

p

-

1

⁢

(

K

)

is compact in

X

.

Title

continuous proper map

Canonical name

ContinuousProperMap

Date of creation

2013-03-22 13:34:22

Last modified on

2013-03-22 13:34:22

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

4

Author

CWoo (3771)

Entry type

Definition

Classification

msc 54-00

Synonym

continuous proper function

Synonym

continuous proper

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

LaTeXML
derived set

derived set

Definition

.

Let

X

be a

topological space

and let

A

be a subset of

X

.
The

derived set

A

′

of

A

is the set of all

limit points

of

A

in

X

[

1

]

.

References

1

J.L. Kelley,

General Topology

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

Title

derived set

Canonical

name

DerivedSet

Date of creation

2013-03-22 13:35:06

Last modified on

2013-03-22 13:35:06

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

7

Author

bwebste (988)

Entry type

Definition

Classification

msc 54-00

Related topic

CantorBendixsonDerivative

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

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diameter

diameter

Let

A

a subset of a

pseudometric space

(

X

,

d

)

. The

diameter

of

A

is defined to be

sup

⁡

{

d

⁢

(

x

,

y

)

:

x

∈

A

,

y

∈

A

}

whenever the supremum exists. If the supremum doesn’t exist, diameter of

A

is defined to be infinite.

Having finite diameter is not a

topological invariant

.

Title

diameter

Canonical name

Diameter

Date of creation

2013-03-22 12:20:36

Last modified on

2013-03-22 12:20:36

Owner

drini (3)

Last modified by

drini (3)

Numerical id

4

Author

drini (3)

Entry type

Definition

Classification

msc 54-00

Related topic

Pi

Generated on Sat Feb 10 11:10:48 2018 by

LaTeXML
discrete space

discrete space

The

discrete topology

on a set

X

is the

topology

given by
the

power set

of

X

. That is, every subset of

X

is open in the discrete topology. A space equipped with the discrete topology is called a

discrete space

.

The discrete topology is the

http://

planetmath

.org/node/3290

finest topology one can give to a set. Any set with the discrete topology is

metrizable

by defining

d

⁢

(

x

,

y

)

=

1

for any

x

,

y

∈

X

with

x

≠

y

, and

d

⁢

(

x

,

x

)

=

0

for any

x

∈

X

.

The following conditions are

equivalent

:

1.

X

is a discrete space.

2.

Every

singleton

in

X

is an

open set

.

3.

Every subset of

X

containing

x

is a

neighborhood

of

x

.

Note that any

bijection

between discrete spaces is trivially a

homeomorphism

.

Discrete Subspaces

If

Y

is a subset of

X

, and the

subspace topology

on

Y

is

discrete

, then

Y

is called a

discrete subspace

or

discrete subset

of

X

.

Suppose

X

is a

topological space

and

Y

is a subset of

X

. Then

Y

is a discrete subspace if and only if, for any

y

∈

Y

, there is an open

S

⊂

X

such that

S

∩

Y

=

{

y

}

.

Examples

1.

ℤ

, as a

metric space

with the standard distance metric

d

⁢

(

m

,

n

)

=

|

m

-

n

|

, has the discrete topology.

2.

ℤ

, as a

subspace

of

ℝ

or

ℂ

with the

usual topology

, is discrete. But

ℤ

, as a subspace of

ℝ

or

ℂ

with the

trivial topology

, is not discrete.

3.

ℚ

, as a subspace of

ℝ

with the usual topology, is not discrete: any open set containing

q

∈

ℚ

contains the

intersection

U

=

B

⁢

(

q

,

ϵ

)

∩

ℚ

of an open ball around

q

with the

rationals

. By the Archimedean property, there’s a

rational number

between

q

and

q

+

ϵ

in

U

. So

U

can’t contain just

q

: singletons can’t be open.

4.

The set of

unit fractions

F

=

{

1

/

n

∣

n

∈

ℕ

}

, as a subspace of

ℝ

with the usual topology, is discrete. But

F

∪

{

0

}

is not, since any open set containing

0

contains some unit fraction.

5.

The

product

of two discrete spaces is discrete under the product topology. The product of an

infinite

number of discrete spaces is discrete under the

box topology

, but if an infinite number of the spaces have more than one

element

, it is not discrete under the product topology.

Title

discrete space

Canonical

name

DiscreteSpace

Date of creation

2013-03-22 12:29:56

Last modified on

2013-03-22 12:29:56

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

17

Author

mathcam (2727)

Entry type

Definition

Classification

msc 54-00

Synonym

discrete topological space

Related topic

Discrete2

Defines

discrete subspace

Defines

discrete topology

Defines

discrete space

Defines

discrete subset

Generated on Sat Feb 10 11:10:50 2018 by

LaTeXML
every second countable space is separable

every second countable space is separable

Theorem 1

.

[

1

]

Every

second countable space

is

separable

.

Proof.

Let

X

be a second countable space and let

ℬ

be a countable base.
For every non-empty set

B

in

ℬ

, choose a point

x

B

∈

B

. The set

A

of all such points

x

B

is clearly

countable

and it’s also dense
since any

open set

intersects

it and thus the whole space is the

closure

of

A

.
That is,

A

is a countably

dense subset

of

X

. Therefore,

X

is separable.
∎

References

1

J.L. Kelley,

General Topology

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

Title

every second countable space is separable

Canonical name

EverySecondCountableSpaceIsSeparable

Date of creation

2013-03-22 12:22:10

Last modified on

2013-03-22 12:22:10

Owner

drini (3)

Last modified by

drini (3)

Numerical id

5

Author

drini (3)

Entry type

Proof

Classification

msc 54-00

Related topic

SecondCountable

Related topic

Separable

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

LaTeXML
finite and countable discrete spaces

finite and countable discrete spaces

Theorem 1

.

Suppose

X

≠

∅

is equipped with the

discrete topology

.

1.

If

X

if finite, then

X

is

homeomorphic

to

{

1

,

…

,

n

}

for some

n

≥

1

.

2.

If

X

if countable, then

X

is homeomorphic to

ℤ

.

Here,

{

1

,

…

,

n

}

and

Z

are endowed with the discrete topology
(or, equivalently, the

subspace topology

from

R

).

Proof.

The first claim will be proven. If

X

=

{

a

1

,

…

,

a

n

}

let

Φ

:

{

1

,

…

,

n

}

→

X

be

Φ

⁢

(

i

)

=

a

i

,

i

=

1

,

…

,

n

.

Since

Φ

is a bijection, it is a

homeomorphism

.

The proof of the second claim is to that of the first.
∎

Title

finite and countable discrete spaces

Canonical name

FiniteAndCountableDiscreteSpaces

Date of creation

2013-03-22 15:17:11

Last modified on

2013-03-22 15:17:11

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

9

Author

matte (1858)

Entry type

Theorem

Classification

msc 54-00

Generated on Sat Feb 10 11:10:54 2018 by

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first axiom of countability

first axiom of countability

A

topological space

(

X

,

τ

)

satisfies the

first axiom of countability

if the

neighborhood system

of every point

x

∈

X

has a countable

local base

. A space that satisfies the first axiom of countability is said to be

first-countable

.

Title

first axiom of countability

Canonical name

FirstAxiomOfCountability

Date of creation

2013-03-22 12:22:13

Last modified on

2013-03-22 12:22:13

Owner

drini (3)

Last modified by

drini (3)

Numerical id

5

Author

drini (3)

Entry type

Definition

Classification

msc 54-00

Synonym

first-countable

Related topic

SecondCountable

Related topic

LocalBase

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

LaTeXML
hemicompact space

hemicompact space

A

topological space

(

X

,

τ

)

is called a

hemicompact

space if there is an

admissible

sequence

in

X

, i.e. there is a sequence of

compact sets

(

K

n

)

n

∈

ℕ

in

X

such that for every

K

⊂

X

compact

there is an

n

∈

ℕ

with

K

⊂

K

n

.

•

The above conditions imply that if

X

is hemicompact with admissible sequence

(

K

n

)

n

∈

ℕ

then

X

=

⋃

n

∈

ℕ

K

n

because every point of

X

is compact and lies in one of the

K

n

.

•

A

hemicompact space

is clearly

σ

-compact. The

converse

is false in general. This follows from the fact that a

first countable

hemicompact space is

locally compact

(see below). Consider the set of rational numbers

ℚ

with the

induced

euclidean topology

.

ℚ

is

σ

-compact but not hemicompact. Since

ℚ

satisfies

the

first axiom of countability

it can’t be hemicompact as this would imply local compactness.

•

Not every locally compact space (like

ℝ

) is hemicompact. Take for example an

uncountable

discrete space

. If we assume in

addition

σ

-

compactness

we obtain a hemicompact space (see below).

Proposition

.

Let

(

X

,

τ

)

be a first countable hemicompact space. Then

X

is locally compact.

Proof.

Let

⋯

⊂

K

n

⊂

K

n

+

1

⊂

⋯

be an admissible sequence of

X

.
Assume for

contradiction

that there is an

x

∈

X

without compact

neighborhood

. Let

U

n

⊃

U

n

+

1

⊃

⋯

be a countable basis for the neighbourhoods of

x

. For every

n

∈

ℕ

choose a point

x

n

∈

U

n

∖

K

n

. The set

K

:=

{

x

n

:

n

∈

ℕ

}

∪

{

x

}

is compact but there is no

n

∈

ℕ

with

K

⊂

K

n

. We have a contradiction.
∎

Proposition.

Let

(

X

,

τ

)

be a locally compact and

σ

-

compact space

. Then

X

is hemicompact.

Proof.

By local compactness we choose a cover

X

⊂

⋃

i

∈

I

U

i

of

open sets

with compact closure (take a compact neighborhood of every point). By

σ

-compactness there is a sequence

(

K

n

)

n

∈

ℕ

of compacts such that

X

=

⋃

n

∈

ℕ

K

n

. To each

K

n

there is a finite subfamily of

(

U

i

)

i

∈

I

which covers

K

n

.
Denote the union of this finite family by

U

n

for each

n

∈

ℕ

. Set

K

~

n

:=

⋃

k

=

1

n

U

k

¯

. Then

(

K

~

n

)

n

∈

ℕ

is a sequence of compacts. Let

K

⊂

X

be compact then there is a finite subfamily of

(

U

i

)

i

∈

I

covering

K

. Therefore

K

⊂

K

n

for some

n

∈

ℕ

.
∎

Title

hemicompact space

Canonical name

HemicompactSpace

Date of creation

2013-03-22 19:08:18

Last modified on

2013-03-22 19:08:18

Owner

karstenb (16623)

Last modified by

karstenb (16623)

Numerical id

8

Author

karstenb (16623)

Entry type

Definition

Classification

msc 54-00

Related topic

SigmaCompact

Defines

hemicompact space

Generated on Sat Feb 10 11:10:57 2018 by

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homotopy groups

homotopy groups

The

homotopy groups

are an infinite series of (covariant)

functors

π

n

indexed by

non-negative integers from

based topological spaces

to groups for

n

>

0

and sets for

n

=

0

.

π

n

⁢

(

X

,

x

0

)

as a set is the set of all

homotopy classes

of maps of pairs

(

D

n

,

∂

⁡

D

n

)

→

(

X

,

x

0

)

, that is, maps of the disk into

X

, taking the

boundary

to the point

x

0

. Alternatively, these can be thought of as maps from the sphere

S

n

into

X

, taking a

basepoint

on the sphere
to

x

0

. These sets are given a group

structure

by declaring the

product

of 2 maps

f

,

g

to simply attaching two disks

D

1

,

D

2

with the right

orientation

along part of their boundaries to get a new disk

D

1

∪

D

2

, and

mapping

D

1

by

f

and

D

2

by

g

, to get a map of

D

1

∪

D

2

. This is

continuous

because we required that the boundary go to a , and well defined up to

homotopy

.

If

f

:

X

→

Y

satisfies

f

⁢

(

x

0

)

=

y

0

, then we get a

homomorphism

of homotopy groups

f

*

:

π

n

⁢

(

X

,

x

0

)

→

π

n

⁢

(

Y

,

y

0

)

by simply composing with

f

. If

g

is a map

D

n

→

X

, then

f

*

⁢

(

[

g

]

)

=

[

f

∘

g

]

.

More algebraically, we can define homotopy groups inductively by

π

n

⁢

(

X

,

x

0

)

≅

π

n

-

1

⁢

(

Ω

⁢

X

,

y

0

)

, where

Ω

⁢

X

is the

loop space

of

X

, and

y

0

is the

constant

path sitting at

x

0

.

If

n

>

1

, the groups we get are abelian.

Homotopy groups are

invariant

under

homotopy equivalence

, and higher homotopy groups (

n

>

1

)
are not changed by the taking of covering spaces.

Some examples are:

π

n

⁢

(

S

n

)

=

ℤ

.

π

m

⁢

(

S

n

)

=

0

if

m

<

n

.

π

n

⁢

(

S

1

)

=

0

if

n

>

1

.

π

n

⁢

(

M

)

=

0

for

n

>

1

where

M

is any surface of nonpositive

Euler characteristic

(not a sphere or

projective plane

).

Title

homotopy groups

Canonical name

HomotopyGroups

Date of creation

2013-03-22 12:15:28

Last modified on

2013-03-22 12:15:28

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

13

Author

bwebste (988)

Entry type

Definition

Classification

msc 54-00

Synonym

higher homotopy groups

Related topic

EilenbergMacLaneSpace

Related topic

HomotopyDoubleGroupoidOfAHausdorffSpace

Related topic

QuantumFundamentalGroupoids

Related topic

CohomologyGroupTheorem

Generated on Sat Feb 10 11:11:00 2018 by

LaTeXML
indiscrete topology

indiscrete topology

If

X

is a set and it is endowed with a

topology

defined by

τ

=

{

X

,

∅

}

then

X

is said to have the

indiscrete topology

.

Furthermore

τ

is the coarsest topology a set can possess, since

τ

would be a subset of any other possible topology. This topology
gives

X

many properties:

•

Every subset of

X

is

sequentially compact

.

•

Every function to a space with the indiscrete topology is

continuous

.

•

X

is

path connected

and hence

connected

but is arc connected only if

X

is uncountable or if

X

has at most a single point. However,

X

is both

hyperconnected

and

ultraconnected

.

•

If

X

has more than one point, it is not

metrizable

because it is not

Hausdorff

. However it is

pseudometrizable

with the metric

d

⁢

(

x

,

y

)

=

0

.

Title

indiscrete topology

Canonical name

IndiscreteTopology

Date of creation

2013-03-22 12:48:11

Last modified on

2013-03-22 12:48:11

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

20

Author

mathwizard (128)

Entry type

Definition

Classification

msc 54-00

Synonym

trivial topology

Synonym

coarse topology

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

LaTeXML
interior

interior

Let

A

be a subset of a

topological space

X

.

The union of all

open sets

contained

in

A

is defined to be the

interior

of

A

.
Equivalently, one could define the interior
of

A

to the be the largest open set contained in

A

.

In this entry we denote the interior of

A

by

int

⁡

(

A

)

.
Another common notation is

A

∘

.

The

exterior

of

A

is defined as
the union of all open sets whose

intersection

with

A

is empty.
That is, the exterior of

A

is the interior of the

complement

of

A

.

The interior of a set enjoys many special

properties

,
some of which are listed below:

1.

int

⁡

(

A

)

⊆

A

2.

int

⁡

(

A

)

is open

3.

int

⁡

(

int

⁡

(

A

)

)

=

int

⁡

(

A

)

4.

int

⁡

(

X

)

=

X

5.

int

⁡

(

∅

)

=

∅

6.

A

is open if and only if

A

=

int

⁡

(

A

)

7.

A

∁

¯

=

(

int

⁡

(

A

)

)

∁

8.

A

¯

∁

=

int

⁡

(

A

∁

)

9.

A

⊆

B

implies that

int

⁡

(

A

)

⊆

int

⁡

(

B

)

10.

int

⁡

(

A

)

=

A

∖

∂

⁡

A

,
where

∂

⁡

A

is the

boundary

of

A

11.

X

=

int

⁡

(

A

)

∪

∂

⁡

A

∪

int

⁡

(

A

∁

)

References

1

S. Willard,

General Topology

,
Addison-Wesley Publishing Company, 1970.

Title

interior

Canonical

name

Interior

Date of creation

2013-03-22 12:48:20

Last modified on

2013-03-22 12:48:20

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

19

Author

yark (2760)

Entry type

Definition

Classification

msc 54-00

Related topic

Complement

Related topic

Closure

Related topic

BoundaryInTopology

Defines

exterior

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

LaTeXML