aarjaneiro/planetmath huggingface dataset:
door space

door space

A

topological space

X

is called a

door space

if every subset of

X

is either open or closed.

From the

definition

, it is immediately clear that any

discrete space

is door.

To find more examples, let us look at the

singletons

of a door space

X

. For each

x

∈

X

, either

{

x

}

is open or closed. Call a point

x

in

X

open or closed according to whether

{

x

}

is open or closed. Let

A

be the

collection

of open points in

X

. If

A

=

X

, then

X

is

discrete

. So suppose now that

A

≠

X

. We look at the special case when

X

-

A

=

{

x

}

. It is now easy to see that the

topology

τ

generated by all the open singletons makes

X

a door space:

Proof.

If

B

⊆

X

does not contain

x

, it is the union of

elements

in

A

, and therefore open. If

x

∈

B

, then its

complement

B

c

does not, so is open, and therefore

B

is closed.
∎

Since

τ

=

P

⁢

(

A

)

∪

{

X

}

, the space

X

not discrete. In

addition

,

X

and

∅

are the only

clopen sets

in

X

.

When

X

-

A

has more than one element, the situation is a little more complicated. We know that if

X

is door, then its topology

𝒯

is

strictly finer

then the topology

τ

generated by all the open singletons. McCartan has shown that

𝒯

=

τ

∪

𝒰

for some

ultrafilter

in

X

. In fact, McCartan showed

𝒯

, as well as the previous two examples, are the only types of possible topologies on a set making it a door space.

References

1

J.L. Kelley,

General Topology

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

2

S.D. McCartan,

Door Spaces are identifiable

, Proc. Roy. Irish Acad. Sect. A, 87 (1) 1987, pp. 13-16.

Title

door space

Canonical

name

DoorSpace

Date of creation

2013-03-22 18:46:11

Last modified on

2013-03-22 18:46:11

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 54E99

Generated on Sat Feb 10 11:29:52 2018 by

LaTeXML
first countable implies compactly generated

first countable implies compactly generated

Proposition 1

.

Any

first countable

topological space

is

compactly generated

.

Proof.

Suppose

X

is first countable, and

A

⊆

X

has the property that, if

C

is any

compact set

in

X

, the set

A

∩

C

is closed in

C

. We want to show tht

A

is closed in

X

. Since

X

is first countable, this is

equivalent

to showing that any

sequence

(

x

i

)

in

A

converging to

x

implies that

x

∈

A

. Let

C

=

{

x

i

∣

i

=

1

,

2

,

…

}

∪

{

x

}

.

Lemma 1

.

C

is

compact

.

Proof.

Let

{

U

j

∣

j

∈

J

}

be a

collection

of

open sets

covering

C

. So

x

∈

U

j

for some

j

. Since

U

j

is open, there is a positive integer

k

such that

x

i

∈

U

j

for all

i

≥

k

. Now, each

x

i

∈

U

d

⁢

(

i

)

for

i

=

1

,

…

,

k

. So

C

is covered by

U

d

⁢

(

1

)

,

…

,

U

d

⁢

(

k

)

, and

U

j

, showing that

C

is compact.
∎

In

addition

, as a

subspace

of

X

,

C

is also first countable. By

assumption

,

A

∩

C

is closed in

C

. Since

x

i

∈

A

∩

C

for all

i

≥

1

, we see that

x

∈

A

∩

C

as well, since

C

is first countable. Hence

x

∈

A

, and

A

is closed in

X

.
∎

Title

first countable implies compactly generated

Canonical name

FirstCountableImpliesCompactlyGenerated

Date of creation

2013-03-22 19:09:35

Last modified on

2013-03-22 19:09:35

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

4

Author

CWoo (3771)

Entry type

Example

Classification

msc 54E99

Generated on Sat Feb 10 11:30:00 2018 by

LaTeXML
Hahn-Mazurkiewicz theorem

Hahn-Mazurkiewicz theorem

A

Hausdorff space

is the

continuous

image of the unit interval

[

0

,

1

]

if and only if it is a

Peano space

.

Title

Hahn-Mazurkiewicz theorem

Canonical name

HahnMazurkiewiczTheorem

Date of creation

2013-03-22 12:59:38

Last modified on

2013-03-22 12:59:38

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 54E99

Related topic

RepresentationTheoremForCompactMetricSpaces

Generated on Sat Feb 10 11:30:02 2018 by

LaTeXML
hyperbolic metric space

hyperbolic metric space

Let

δ

≥

0

. A

metric space

(

X

,

d

)

is

δ

hyperbolic

if, for any figure

A

⁢

B

⁢

C

in

X

that is a

geodesic triangle

with respect to

d

and for every

P

∈

A

⁢

B

¯

, there exists a point

Q

∈

A

⁢

C

¯

∪

B

⁢

C

¯

such that

d

⁢

(

P

,

Q

)

≤

δ

.

A

hyperbolic metric space

is a metric space that is

δ

hyperbolic for some

δ

≥

0

.

Although a metric space is hyperbolic if it is

δ

hyperbolic for some

δ

≥

0

, one usually tries to find the smallest value of

δ

for which a hyperbolic metric space

(

X

,

d

)

is

δ

hyperbolic.

A example of a hyperbolic metric space is the

real line

under the usual metric. Given any three points

A

,

B

,

C

∈

ℝ

, we always have that

A

⁢

B

¯

⊆

A

⁢

C

¯

∪

B

⁢

C

¯

. Thus, for any

P

∈

A

⁢

B

¯

, we can take

Q

=

P

. Therefore, the real line is 0 hyperbolic. reasoning can be used to show that every

real tree

is 0 hyperbolic.

Title

hyperbolic metric space

Canonical

name

HyperbolicMetricSpace

Date of creation

2013-03-22 17:11:29

Last modified on

2013-03-22 17:11:29

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

6

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 54E99

Classification

msc 54E35

Classification

msc 20F06

Defines

δ

hyperbolic

Generated on Sat Feb 10 11:30:04 2018 by

LaTeXML
real tree

real tree

A

metric space

X

is said to be a

real tree

or

R

-tree

, if for each

x

,

y

∈

X

there is a unique arc
from

x

to

y

, and furthermore this arc is an

isometric

http://planetmath.org/node/429

embedding

.

Every real tree is a

hyperbolic metric space

; moreover, every real tree is 0 hyperbolic.

The

Cayley graph

of any

free group

is considered to be a real tree. Note that its graph is a tree in the graph theoretic sense. To make it a real tree, we view the edges as isometric (

http://planetmath.org/Isometric

) to the

line segment

[

0

,

1

]

under a (

surjective

) isometry (

http://planetmath.org/Isometry

) and attach the edges to the tree. The resulting 1-complex is then a

locally finite

real tree. Because of this result, every free group is a

hyperbolic group

.

Title

real tree

Canonical name

RealTree

Date of creation

2013-03-22 15:16:55

Last modified on

2013-03-22 15:16:55

Owner

GrafZahl (9234)

Last modified by

GrafZahl (9234)

Numerical id

10

Author

GrafZahl (9234)

Entry type

Definition

Classification

msc 54E99

Classification

msc 54E40

Synonym

ℝ

-tree

Related topic

MetricSpace

Related topic

Arc

Related topic

Curve

Related topic

SNCFMetric

Related topic

Isometry

Related topic

FreeGroup

Related topic

HyperbolicGroup

Generated on Sat Feb 10 11:30:19 2018 by

LaTeXML
sober space

sober space

Let

X

be a

topological space

. A subset

A

of

X

is said to be

irreducible

if whenever

A

⊆

B

∪

C

with

B

,

C

closed, we have

A

⊆

B

or

A

⊆

C

. Any

singleton

and its

closure

are irreducible. More generally, the closure of an

irreducible set

is irreducible.

A topological space

X

is called a

sober space

if every irreducible

closed subset

is the closure of some

unique

point in

X

.

Remarks

.

•

For any sober space, the closure of a point determines the point. In other words,

cl

⁡

(

x

)

=

cl

⁡

(

y

)

implies

x

=

y

.

•

A space is sober iff the closure of every irreducible set is the closure of a unique point.

•

Any sober space is T0.

•

Any

Hausdorff space

is sober.

•

A closed subspace of a sober space is sober.

•

Any

product

of sober spaces is sober.

Title

sober space

Canonical

name

SoberSpace

Date of creation

2013-03-22 16:43:44

Last modified on

2013-03-22 16:43:44

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 54E99

Defines

irreducible set

Generated on Sat Feb 10 11:30:28 2018 by

LaTeXML
Vitali covering

Vitali covering

A

collection

of sets

𝒱

in a

metric space

X

is called a

Vitali covering

(or Vitali class) for

X

if for each

x

∈

X

and

δ

>

0

there exists

U

∈

𝒱

such that

x

∈

U

and

0

<

diam

⁡

(

U

)

≤

δ

.

Title

Vitali covering

Canonical name

VitaliCovering

Date of creation

2013-03-22 13:29:48

Last modified on

2013-03-22 13:29:48

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

6

Author

Koro (127)

Entry type

Definition

Classification

msc 54E99

Defines

Vitali class

Generated on Sat Feb 10 11:30:29 2018 by

LaTeXML
wedge product of pointed topological spaces

wedge product of pointed topological spaces

Definition

.

Let

{

(

X

i

,

x

i

)

}

i

∈

I

be a finite family of disjoint

pointed topological spaces

. The

wedge product

of these spaces is

⋁

i

∈

I

X

i

=

(

⋃

i

∈

I

X

i

)

/

{

x

i

:

i

∈

I

}

.

This can be generalized to arbitrary families of pointed topological spaces, although this may require that the

topology

on

⋃

i

∈

I

X

i

satisfy a coherence condition (see

[

1

]

).

References

1

Munkres, J. R. (2000).

Topology

(2nd. ed.). Upper Saddle River, NJ: Prentice Hall.

2

Prasolov, V. V. (2004).

Elements of combinatorial and

differential topology

. Providence, RI:

American Mathematical Society

.

3

Shick, P. L. (2007).

Topology: Point-set and geometric

. Hoboken, NJ: John Wiley & Sons.

Title

wedge product of pointed topological spaces

Canonical

name

WedgeProductOfPointedTopologicalSpaces

Date of creation

2013-03-22 18:49:09

Last modified on

2013-03-22 18:49:09

Owner

MichaelMcCliment (20205)

Last modified by

MichaelMcCliment (20205)

Numerical id

5

Author

MichaelMcCliment (20205)

Entry type

Definition

Classification

msc 54E99

Synonym

wedge

Synonym

wedge product

Related topic

QuotientSpace

Related topic

CategoryOfPointedTopologicalSpaces

Generated on Sat Feb 10 11:30:31 2018 by

LaTeXML
continuum

continuum

A

continuum

is a

compact

connected

topological space

.
Some authors impose additional conditions and require that the space be nondegenerate,

Hausdorff

,
or metric.

References

SS

Lynn Arthur Steen and J. Arthur Seebach, Jr,

Counterexamples in Topology

, Springer-Verlag, 1978, p. 33

HY

John G. Hocking, and Gail S. Young,

Topology

, Dover Publications, New York, 1988, p. 43

G

Steven A. Gaal,

Point Set Topology

, Academic Press, New York, 1964, p. 103

MCG

Michael C. Gemignani,

Elementary

Topology, 2nd ed.

Dover Publications, New York, 1990, p. 202

NEW

M.H.A. Newman,

Elements of the Topology of Plane Sets of Points

, Cambridge University Press, 1964, p. 71

W

Stephen Willard,

General Topology

, Addison-Wesley, Reading, MA, 1970, p. 203

WI

Raymond Louis Wilder,

Topology of Manifolds

, Amer. Math. Society, Providence, RI, 1963, p. 36

MI

J. van Mill, G.M. Reed, editors,

Open Problems

in Topology

, North-Holland, Amsterdam, 1990, p. 305

Title

continuum

Canonical name

Continuum

Date of creation

2013-03-22 18:37:22

Last modified on

2013-03-22 18:37:22

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

5

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 54F15

Generated on Sat Feb 10 11:30:42 2018 by

LaTeXML
partial ordering in a topological space

partial ordering in a topological space

Let

X

be a

topological space

. For any

x

,

y

∈

X

, we define a

binary relation

≤

on

X

as follows:

x

≤

y

iff

x

∈

{

y

}

¯

.

Proposition 1

.

≤

is a

preorder

.

Proof.

Clearly

x

≤

x

. Next, suppose

x

≤

y

and

y

≤

z

. Let

C

be a

closed set

containing

z

. Since

y

is in the

closure

of

{

z

}

,

y

∈

C

. Since

x

is in the closure of

{

y

}

,

x

∈

C

also. So

x

≤

z

. ∎

We call

≤

the

specialization preorder

on

X

. If

x

≤

y

, then

x

is called a

specialization point

of

y

, and

y

a

generization point

of

x

. For any set

A

⊆

X

,

•

the set of all specialization points of points of

A

is called the

specialization

of

A

, and is denoted by

Sp

⁢

(

A

)

;

•

the set of all generization points of points of

A

is called the

generization

of

A

, and is denoted by

Gen

⁢

(

A

)

.

Proposition 2

.

. If

X

is

T

0

(

http://planetmath.org/T0

), then

≤

is a

partial order

.

Proof.

Suppose next that

x

≤

y

and

y

≤

x

. If

x

≠

y

, then there is an

open set

A

such that

x

∈

A

and

y

∉

A

. So

y

∈

A

c

, the

complement

of

A

, which is a closed set. But then

x

∈

A

c

since it is in the closure of

{

y

}

. So

x

∈

A

∩

A

c

=

∅

, a contradition. Thus

x

=

y

. ∎

This turns a

T

0

topological space into a poset, where

≤

here is called the

specialization order

of the space.

Given a

T

0

space, we have the following:

Proposition 3

.

x

≤

y

iff

x

∈

U

implies

y

∈

U

for any open set

U

in

X

.

Proof.

(

⇒

)

:

if

x

∈

U

and

y

∉

U

, then

y

∈

U

c

. Since

x

≤

y

, we have

x

∈

U

c

, a

contradiction

.

(

⇐

)

:

if

x

∉

{

y

}

¯

, then for some closed set

C

, we have

y

∈

C

and

x

∉

C

. But then

x

∈

C

c

, so that

y

∈

C

c

, a contradiction. ∎

Remarks

.

•

{

x

}

¯

=

↓

x

, the

lower set

of

x

. (

z

∈

↓

x

iff

z

≤

x

iff

z

∈

{

x

}

¯

).

•

But if

X

is

T

1

(

http://planetmath.org/T1

), then the partial ordering just defined is

trivial

(the diagonal set), since every point is a

closed point

(for verification, just modify the antisymmetry portion of the above proof).

Title

partial ordering in a topological space

Canonical name

PartialOrderingInATopologicalSpace

Date of creation

2013-03-22 16:35:02

Last modified on

2013-03-22 16:35:02

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 54F99

Defines

specialization order

Defines

specialization preorder

Defines

specialization

Defines

generization

Generated on Sat Feb 10 11:30:53 2018 by

LaTeXML
the real numbers are indecomposable as a topological space

the real numbers are indecomposable as a topological space

Let

ℝ

be the set of real numbers with

standard topology

. We wish to show that if

ℝ

is

homeomorphic

to

X

×

Y

for some

topological spaces

X

and

Y

, then either

X

is one point space or

Y

is one point space. First let us prove a lemma:

Lemma

. Let

X

and

Y

be

path connected

topological spaces such that

cardinality

of both

X

and

Y

is at least

2

. Then for any point

(

x

0

,

y

0

)

∈

X

×

Y

the space

X

×

Y

∖

{

(

x

0

,

y

0

)

}

with

subspace topology

is path connected.

Proof

. Let

x

′

∈

X

and

y

′

∈

Y

such that

x

′

≠

x

0

and

y

′

≠

y

0

(we assumed that such points exist). It is

sufficient

to show that for any point

(

x

1

,

y

1

)

from

X

×

Y

∖

{

(

x

0

,

y

0

)

}

there exists a continous map

σ

:

I

→

X

×

Y

such that

σ

⁢

(

0

)

=

(

x

1

,

y

1

)

,

σ

⁢

(

1

)

=

(

x

′

,

y

′

)

and

(

x

0

,

y

0

)

∉

σ

⁢

(

I

)

.

Let

(

x

1

,

y

1

)

∈

X

×

Y

∖

{

(

x

0

,

y

0

)

}

. Therefore either

x

1

≠

x

0

or

y

1

≠

y

0

. Assume that

y

1

≠

y

0

(the other case is analogous). Choose paths

σ

:

I

→

X

from

x

1

to

x

′

and

τ

:

I

→

Y

from

y

1

to

y

′

. Then we have induced paths:

σ

′

:

I

→

X

×

Y

such

⁢

that

⁢

σ

′

⁢

(

t

)

=

(

σ

⁢

(

t

)

,

y

1

)

;

τ

′

:

I

→

X

×

Y

such

⁢

that

⁢

τ

′

⁢

(

t

)

=

(

x

′

,

τ

⁢

(

t

)

)

.

Then the path

σ

′

*

τ

′

:

I

→

X

×

Y

defined by the

formula

(

σ

′

*

τ

′

)

⁢

(

t

)

=

{

σ

′

⁢

(

2

⁢

t

)

when

⁢

0

≤

t

≤

1

2

τ

′

⁢

(

2

⁢

t

-

1

)

when

1

2

≤

t

≤

1

is a desired path.

□

Proposition

. If there exist topological spaces

X

and

Y

such that

ℝ

is homeomorphic to

X

×

Y

, then either

X

has exactly one point or

Y

has exactly one point.

Proof

. Assume that neither

X

nor

Y

has exactly one point. Now

X

×

Y

is path connected since it is homeomorphic to

ℝ

, so it is well known that both

X

and

Y

have to be path connected (please see this entry (

http://planetmath.org/ProductOfPathConnectedSpacesIsPathConnected

) for more details). Therefore for any point

(

x

,

y

)

∈

X

×

Y

the space

X

×

Y

∖

{

(

x

,

y

)

}

is also path connected (due to lemma), but there exists a

real number

r

∈

ℝ

such that

X

×

Y

∖

{

(

x

,

y

)

}

is homeomorphic to

ℝ

∖

{

r

}

.

Contradiction

, since

ℝ

∖

{

r

}

is not path connected.

□

Title

the real numbers are

indecomposable

as a topological space

Canonical name

TheRealNumbersAreIndecomposableAsATopologicalSpace

Date of creation

2013-03-22 18:30:59

Last modified on

2013-03-22 18:30:59

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

12

Author

joking (16130)

Entry type

Theorem

Classification

msc 54F99

Generated on Sat Feb 10 11:30:56 2018 by

LaTeXML
the sphere is indecomposable as a topological space

the sphere is indecomposable as a topological space

Proposition

. If for any

topological spaces

X

and

Y

the

n

-dimensional sphere

𝕊

n

is

homeomorphic

to

X

×

Y

, then either

X

has exactly one point or

Y

has exactly one point.

Proof

. Recall that the

homotopy group

functor is additive, i.e.

π

n

⁢

(

X

×

Y

)

≃

π

n

⁢

(

X

)

⊕

π

n

⁢

(

Y

)

. Assume that

𝕊

n

is homeomorphic to

X

×

Y

. Now

π

n

⁢

(

𝕊

n

)

≃

ℤ

and thus we have:

ℤ

≃

π

n

⁢

(

𝕊

n

)

≃

π

n

⁢

(

X

×

Y

)

≃

π

n

⁢

(

X

)

⊕

π

n

⁢

(

Y

)

.

Since

ℤ

is an

indecomposable group

, then either

π

n

⁢

(

X

)

≃

0

or

π

n

⁢

(

Y

)

≃

0

.

Assume that

π

n

⁢

(

Y

)

≃

0

. Consider the map

p

:

X

×

Y

→

Y

such that

p

⁢

(

x

,

y

)

=

y

. Since

X

×

Y

is homeomorphic to

𝕊

n

and

π

n

⁢

(

Y

)

≃

0

, then

p

is

homotopic

to some constant map. Let

y

0

∈

Y

and

H

:

I

×

X

×

Y

→

Y

be such that

H

⁢

(

0

,

x

,

y

)

=

p

⁢

(

x

,

y

)

=

y

;

H

⁢

(

1

,

x

,

y

)

=

y

0

.

Consider the map

F

:

I

×

X

×

Y

→

X

×

Y

defined by the formula

F

⁢

(

t

,

x

,

y

)

=

(

x

,

H

⁢

(

t

,

x

,

y

)

)

.

Note that

F

⁢

(

0

,

x

,

y

)

=

(

x

,

y

)

and

F

⁢

(

1

,

x

,

y

)

=

(

x

,

y

0

)

and thus

X

×

{

y

0

}

is a

deformation retract

of

X

×

Y

. But

X

×

Y

is a sphere and spheres do not have proper deformation retracts (please see this entry (

http://planetmath.org/EveryMapIntoSphereWhichIsNotOntoIsNullhomotopic

) for more details). Therefore

X

×

{

y

0

}

=

X

×

Y

, so

Y

=

{

y

0

}

has exactly one point.

□

Title

the sphere is indecomposable as a topological space

Canonical

name

TheSphereIsIndecomposableAsATopologicalSpace

Date of creation

2013-03-22 18:31:45

Last modified on

2013-03-22 18:31:45

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

8

Author

joking (16130)

Entry type

Theorem

Classification

msc 54F99

Generated on Sat Feb 10 11:30:58 2018 by

LaTeXML
compact Hausdorff space is extremally disconnected if its function algebra is a bounded complete lattice

compact Hausdorff space is extremally disconnected if its function algebra is a bounded complete lattice

Let

X

be a

compact

Hausdorff space

and

C

⁢

(

X

)

the algebra of

continuous functions

X

⟶

ℂ

. Recall that

C

⁢

(

X

)

is a

vector lattice

with the usual order (

http://planetmath.org/PartialOrder

):

f

≤

g

⟺

g

-

f

takes

positive

(or zero) values.

Theorem

-

If every subset of

C

⁢

(

X

)

that is

bounded from above

has a least

upper bound

(i.e.

C

⁢

(

X

)

is a

bounded complete

lattice

), then

X

is

extremally disconnected

.

Title

compact Hausdorff space is extremally disconnected if its

function

algebra is a bounded complete lattice

Canonical name

CompactHausdorffSpaceIsExtremallyDisconnectedIfItsFunctionAlgebraIsABoundedCompleteLattice

Date of creation

2013-03-22 17:53:08

Last modified on

2013-03-22 17:53:08

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

5

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 54G05

Classification

msc 46J10

Classification

msc 06F20

Synonym

sufficient

condition for a compact Hausdorf space to be extremally disconnected

Generated on Sat Feb 10 11:31:00 2018 by

LaTeXML
extremally disconnected

extremally disconnected

A

topological space

X

is said to be

extremally disconnected

if every

open set

in

X

has an open

closure

.

It can be shown that

X

is extremally disconnected iff any two disjoint open sets in

X

have disjoint closures. Every extremally disconnected space is

totally disconnected

.

Notes

Some authors like

[

1

]

and

[

2

]

use the above
definition as is, while others (e.g.

[

3

,

4

]

) require
that an extremally disconnected space should (in addition to the above
condition) also be a

Hausdorff space

.

References

1

S. Willard,

General Topology

,
Addison-Wesley, Publishing Company, 1970.

2

J. L. Kelley,

General Topology

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

3

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

Counterexamples in topology

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

4

N. Bourbaki,

General Topology, Part 1

,
Addison-Wesley Publishing Company, 1966.

Title

extremally disconnected

Canonical name

ExtremallyDisconnected

Date of creation

2013-03-22 12:42:00

Last modified on

2013-03-22 12:42:00

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

8

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 54G05

Synonym

extremely disconnected

Related topic

ConnectedSpace

Generated on Sat Feb 10 11:31:01 2018 by

LaTeXML