aarjaneiro/planetmath huggingface dataset:
product topology preserves the Hausdorff property

product topology preserves the Hausdorff property

Theorem

Suppose

{

X

α

}

α

∈

A

is a

collection

of

Hausdorff spaces

. Then the

generalized Cartesian product

∏

α

∈

A

X

α

equipped with the

product topology

is a Hausdorff space.

Proof.

Let

Y

=

∏

α

∈

A

X

α

, and
let

x

,

y

be distinct points in

Y

. Then there is an index

β

∈

A

such that

x

⁢

(

β

)

and

y

⁢

(

β

)

are distinct points in
the Hausdorff space

X

β

. It follows that there are

open sets

U

and

V

in

X

β

such that

x

⁢

(

β

)

∈

U

,

y

⁢

(

β

)

∈

V

,
and

U

∩

V

=

∅

.
Let

π

β

be the projection operator

Y

→

X

β

defined
here (

http://planetmath.org/GeneralizedCartesianProduct

). By the

definition

of
the product topology,

π

β

is

continuous

, so

π

β

-

1

⁢

(

U

)

and

π

β

-

1

⁢

(

V

)

are open sets in

Y

. Also,
since the

preimage

commutes with set

operations

(

http://planetmath.org/InverseImageCommutesWithSetOperations

),
we have that

π

β

-

1

⁢

(

U

)

∩

π

β

-

1

⁢

(

V

)

=

π

β

-

1

⁢

(

U

∩

V

)

=

∅

.

Finally, since

x

⁢

(

β

)

∈

U

, i.e.,

π

β

⁢

(

x

)

∈

U

,
it follows
that

x

∈

π

β

-

1

⁢

(

U

)

. Similarly,

y

∈

π

β

-

1

⁢

(

V

)

.
We have shown that

U

and

V

are open

disjoint

neighborhoods

of

x

respectively

y

. In other words,

Y

is a Hausdorff space.

□

Title

product topology

preserves

the

Hausdorff

property

Canonical

name

ProductTopologyPreservesTheHausdorffProperty

Date of creation

2013-03-22 13:39:40

Last modified on

2013-03-22 13:39:40

Owner

archibal (4430)

Last modified by

archibal (4430)

Numerical id

7

Author

archibal (4430)

Entry type

Theorem

Classification

msc 54B10

Classification

msc 54D10

Generated on Sat Feb 10 11:17:09 2018 by

LaTeXML
cell attachment

cell attachment

Let

X

be a

topological space

,
and let

Y

be the

adjunction

Y

:=

X

∪

φ

D

k

,
where

D

k

is a closed

k

-ball (

http://planetmath.org/StandardNBall

)
and

φ

:

S

k

-

1

→

X

is a

continuous map

,
with

S

k

-

1

is the

(

k

-

1

)

-sphere considered as the

boundary

of

D

k

.
Then, we say that

Y

is obtained from

X

by the

attachment of a

k

-cell,

by the

attaching map

φ

.

The image

e

k

of

D

k

in

Y

is called a

closed

k

-cell

,
and the image

e

∘

k

of the

interior

D

∘

:=

D

k

∖

S

k

-

1

of

D

k

is the corresponding

open

k

-cell

.

Note that for

k

=

0

the above definition reduces to
the statement that

Y

is the

disjoint union

of

X

with a one-point space.

More generally, we say that

Y

is obtained from

X

by

cell attachment

if

Y

is

homeomorphic

to an adjunction

X

∪

{

φ

i

}

D

k

i

,
where the maps

{

φ

i

}

into

X

are defined on the boundary spheres of

closed balls

{

D

k

i

}

.

Title

cell attachment

Canonical name

CellAttachment

Date of creation

2013-03-22 13:25:53

Last modified on

2013-03-22 13:25:53

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

13

Author

yark (2760)

Entry type

Definition

Classification

msc 54B15

Synonym

cell adjunction

Related topic

CWComplex

Defines

cell

Defines

open cell

Defines

closed cell

Defines

attaching map

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

LaTeXML
feature of Mobius strip

feature of Mobius strip

The Möbius band can be considered as the “twisted”

I

- bundle over the circle, that is, a bundle over

S

1

and fiber the

interval

[

0

,

1

]

which is not the trivial one:

I

×

S

1

.

Title

feature of Mobius strip

Canonical name

FeatureOfMobiusStrip

Date of creation

2013-03-22 15:56:37

Last modified on

2013-03-22 15:56:37

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

8

Author

juanman (12619)

Entry type

Feature

Classification

msc 54B15

Generated on Sat Feb 10 11:17:13 2018 by

LaTeXML
Klein bottle

Klein bottle

Where a Möbius strip is a two dimensional object with only one

surface

and one edge, a

Klein bottle

is a two dimensional object with a single surface, and no edges. Consider for comparison, that a sphere is a two dimensional surface with no edges, but that has two surfaces.

A Klein bottle can be constructed by taking a rectangular subset of

ℝ

2

and identifying opposite edges with each other, in the following fashion:

Consider the rectangular subset

[

-

1

,

1

]

×

[

-

1

,

1

]

. Identify the points

(

x

,

1

)

with

(

x

,

-

1

)

, and the points

(

1

,

y

)

with the points

(

-

1

,

-

y

)

. Doing these two operations simultaneously will give you the Klein bottle.

Visually, the above is accomplished by the following. Take a rectangle, and match up the arrows on the edges so that their

orientation

matches:

This of course is completely impossible to do physically in 3-dimensional space; to be able to properly create a Klein bottle, one would need to be able to build it in 4-dimensional space.

To construct a pseudo-Klein bottle in 3-dimensional space, you would first take a

cylinder

and cut a hole at one point on the side. Next, bend one end of the cylinder through that hole, and attach it to the other end of the clyinder.

A Klein bottle may be parametrized by the following equations:

x

=

{

a

⁢

cos

⁡

(

u

)

⁢

(

1

+

sin

⁡

(

u

)

)

+

r

⁢

cos

⁡

(

u

)

⁢

cos

⁡

(

v

)

0

≤

u

<

π

a

⁢

cos

⁡

(

u

)

⁢

(

1

+

sin

⁡

(

u

)

)

+

r

⁢

cos

⁡

(

v

+

π

)

π

<

u

≤

2

⁢

π

y

=

{

b

⁢

sin

⁡

(

u

)

+

r

⁢

sin

⁡

(

u

)

⁢

cos

⁡

(

v

)

0

≤

u

<

π

b

⁢

sin

⁡

(

u

)

π

<

u

≤

2

⁢

π

z

=

r

⁢

sin

⁡

(

v

)

where

v

∈

[

0

,

2

⁢

π

]

,

u

∈

[

0

,

2

⁢

π

]

,

r

=

c

⁢

(

1

-

cos

⁡

(

u

)

2

)

and

a

,

b

,

c

are chosen arbitrarily.

Title

Klein bottle

Canonical

name

KleinBottle

Date of creation

2013-03-22 13:37:00

Last modified on

2013-03-22 13:37:00

Owner

vernondalhart (2191)

Last modified by

vernondalhart (2191)

Numerical id

12

Author

vernondalhart (2191)

Entry type

Definition

Classification

msc 54B15

Related topic

MobiusStrip

Generated on Sat Feb 10 11:17:15 2018 by

LaTeXML
Möbius strip

Möbius strip

A

Möbius strip

is a non-orientiable 2-dimensional

surface

with a 1-dimensional

boundary

. It can be embedded in

ℝ

3

, but only has a single .

We can parameterize the Möbius strip by

x

=

r

⋅

cos

⁡

θ

,

y

=

r

⋅

sin

⁡

θ

,

z

=

(

r

-

2

)

⁢

tan

⁡

θ

2

.

The Möbius strip is therefore a subset of the solid torus.

Topologically, the Möbius strip is formed by taking a

quotient space

of

I

2

=

[

0

,

1

]

×

[

0

,

1

]

⊂

ℝ

2

. We do this by first letting

M

be the

partition

of

I

2

formed by the

equivalence relation

:

(

1

,

x

)

∼

(

0

,

1

-

x

)

where

0

≤

x

≤

1

,

and every other point in

I

2

is only related to itself.

By giving

M

the quotient topology given by the quotient map

p

:

I

2

→

M

we obtain the Möbius strip.

Schematically we can

represent

this identification as follows:

Diagram

1: The identifications made on

I

2

to make a Möbius strip.

We identify two opposite sides but with different

orientations

.

Since the Möbius strip is

homotopy equivalent

to a circle, it has

ℤ

as its

fundamental group

. It is not however,

homeomorphic

to the circle, although its boundary is.

Title

Möbius strip

Canonical

name

MobiusStrip

Date of creation

2013-03-22 12:55:28

Last modified on

2013-03-22 12:55:28

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

22

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 54B15

Synonym

Möbius band

Related topic

KleinBottle

Related topic

Torus

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

LaTeXML
quotient space

quotient space

Let

X

be a

topological space

, and let

∼

be an

equivalence relation

on

X

. Write

X

*

for the set of

equivalence classes

of

X

under

∼

. The

quotient topology

on

X

*

is the

topology

whose

open sets

are the subsets

U

⊂

X

*

such that

⋃

U

⊂

X

is an open subset of

X

. The space

X

*

is called the

quotient space

of the space

X

with respect to

∼

. It is often written

X

/

∼

.

The

projection map

π

:

X

⟶

X

*

which sends each element of

X

to its equivalence class is always a

continuous map

. In fact, the map

π

satisfies

the

stronger

property

that a subset

U

of

X

*

is open if and only if the subset

π

-

1

⁢

(

U

)

of

X

is open. In general, any surjective map

p

:

X

⟶

Y

that satisfies this stronger property is called a

quotient map

, and given such a quotient map, the space

Y

is always

homeomorphic

to the quotient space of

X

under the equivalence relation

x

∼

x

′

⇔

p

⁢

(

x

)

=

p

⁢

(

x

′

)

.

As a set, the construction of a quotient space collapses each of the equivalence classes of

∼

to a single point. The topology on the quotient space is then chosen to be the strongest topology such that the projection map

π

is

continuous

.

For

A

⊂

X

, one often writes

X

/

A

for the quotient space obtained by identifying all the points of

A

with each other.

Title

quotient space

Canonical

name

QuotientSpace

Date of creation

2013-03-22 12:39:40

Last modified on

2013-03-22 12:39:40

Owner

djao (24)

Last modified by

djao (24)

Numerical id

5

Author

djao (24)

Entry type

Definition

Classification

msc 54B15

Related topic

AdjunctionSpace

Defines

quotient topology

Defines

quotient map

Generated on Sat Feb 10 11:17:19 2018 by

LaTeXML
torus

torus

Visually, the torus looks like a doughnut. Informally, we take a

rectangle

, identify two edges to form a

cylinder

, and then identify the two ends of the cylinder to form the torus. Doing this gives us a surface of genus one. It can also be described as the

Cartesian product

of two circles, that is,

S

1

×

S

1

. The torus can be parameterized in

Cartesian coordinates

by:

x

=

cos

⁡

(

s

)

⋅

(

R

+

r

⋅

cos

⁡

(

t

)

)

y

=

sin

⁡

(

s

)

⋅

(

R

+

r

⋅

cos

⁡

(

t

)

)

z

=

r

⋅

sin

⁡

(

t

)

with

R

the

major radius

and

r

the

minor radius

are

constant

, and

s

,

t

∈

[

0

,

2

⁢

π

)

.

Figure 1: A torus generated with

Mathematica

4.1

To create the torus mathematically, we start with the

closed subset

X

=

[

0

,

1

]

×

[

0

,

1

]

⊆

ℝ

2

. Let

X

*

be the set with

elements

:

{

x

×

0

,

x

×

1

∣

0

<

x

<

1

}

{

0

×

y

,

1

×

y

∣

0

<

y

<

1

}

and also the four-point set

{

0

×

0

,

1

×

0

,

0

×

1

,

1

×

1

}

.

This can be schematically represented in the following diagram.

Diagram 1: The identifications made on

I

2

to make a torus.

Opposite sides are identified with equal

orientations

, and the four corners

are identified to one point.

Note that

X

*

is a

partition

of

X

, where we have identified opposite sides of the square together, and all four corners together. We can then form the

quotient topology

induced

by the quotient map

p

:

X

⟶

X

*

by sending each element

x

∈

X

to the corresponding element of

X

*

containing

x

.

Title

torus

Canonical name

Torus

Date of creation

2013-03-22 12:55:17

Last modified on

2013-03-22 12:55:17

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

15

Author

Daume (40)

Entry type

Definition

Classification

msc 54B15

Classification

msc 51H05

Related topic

MobiusStrip

Related topic

NTorus

Related topic

SurfaceOfRevolution2

Defines

major radius

Defines

minor radius

Generated on Sat Feb 10 11:17:21 2018 by

LaTeXML
adjunction space

adjunction space

Let

X

and

Y

be

topological spaces

, and let

A

be a

subspace

of

Y

. Given a

continuous function

f

:

A

→

X

,

define the space

Z

:=

X

∪

f

Y

to be the

quotient space

X

∐

Y

/

∼

,

where the symbol

∐

stands for

disjoint union

and the

equivalence relation

∼

is generated by

y

∼

f

⁢

(

y

)

for all

y

∈

A

.

Z

is called an

adjunction

of

Y

to

X

along

f

(or along

A

, if the map

f

is understood). This construction has the effect of gluing the subspace

A

of

Y

to its image in

X

under

f

.

Remark 1

Though the definition makes sense for arbitrary

A

, it is usually assumed that

A

is a closed subspace of

Y

. This results in better-behaved adjunction spaces (e.g., the quotient of

X

by a non-closed set is never

Hausdorff

).

Remark 2

The adjunction space construction is a special case of the pushout in the category of topological spaces. The two maps being pushed out are

f

and the

inclusion map

of

A

into

Y

.

Title

adjunction space

Canonical name

AdjunctionSpace

Date of creation

2013-03-22 13:25:56

Last modified on

2013-03-22 13:25:56

Owner

antonio (1116)

Last modified by

antonio (1116)

Numerical id

10

Author

antonio (1116)

Entry type

Definition

Classification

msc 54B17

Related topic

QuotientSpace

Defines

adjunction

Generated on Sat Feb 10 11:17:23 2018 by

LaTeXML
direct image

direct image

If

f

:

X

→

Y

is a

continuous map

of

topological spaces

and

ℱ

is a sheaf on

X

, the

direct image

sheaf,

f

*

⁢

ℱ

on

Y

is defined by

(

f

*

⁢

ℱ

)

⁢

(

V

)

=

ℱ

⁢

(

f

-

1

⁢

(

V

)

)

for

open sets

V

⊂

Y

, with the

restriction

maps induced from those of

ℱ

.

Title

direct image

Canonical name

DirectImage1

Date of creation

2013-03-22 12:03:10

Last modified on

2013-03-22 12:03:10

Owner

nerdy2 (62)

Last modified by

nerdy2 (62)

Numerical id

8

Author

nerdy2 (62)

Entry type

Definition

Classification

msc 54B40

Generated on Sat Feb 10 11:17:25 2018 by

LaTeXML
sheaf

sheaf

1

Presheaves

Let

X

be a

topological space

and let

𝒜

be a

category

. A

presheaf

on

X

with values in

𝒜

is a

contravariant functor

F

from the category

ℬ

whose objects are

open sets

in

X

and whose

morphisms

are

inclusion mappings

of open sets of

X

, to
the category

𝒜

.

As this definition may be less than helpful to many readers, we offer
the following

equivalent

(but longer) definition. A

presheaf

F

on

X

consists of the following data:

1.

An object

F

⁢

(

U

)

in

𝒜

, for each open set

U

⊂

X

2.

A morphism

res

V

,

U

:

F

⁢

(

V

)

→

F

⁢

(

U

)

for each pair of open
sets

U

⊂

V

in

X

(called the

restriction

morphism

), such
that:

(a)

For every open set

U

⊂

X

, the morphism

res

U

,

U

is
the identity morphism.

(b)

For any open sets

U

⊂

V

⊂

W

in

X

, the

diagram

Generated on Sat Feb 10 11:17:30 2018 by

LaTeXML
stalk

stalk

Let

F

be a

presheaf

over a

topological space

X

with values in an

abelian category

𝒜

, and suppose

direct limits

exist in

𝒜

. For any point

p

∈

X

, the

stalk

F

p

of

F

at

p

is defined to be the object in

𝒜

which is the direct limit of the objects

F

⁢

(

U

)

over the

directed set

of all

open sets

U

⊂

X

containing

p

, with respect to the

restriction

morphisms

of

F

. In other words,

F

p

:=

lim

⟶

U

∋

p

⁢

F

⁢

(

U

)

If

𝒜

is a

category

consisting of sets, the stalk

F

p

can be viewed as the set of all germs of

sections

of

F

at the point

p

. That is, the set

F

p

consists of all the

equivalence classes

of

ordered pairs

(

U

,

s

)

where

p

∈

U

and

s

∈

F

⁢

(

U

)

, under the

equivalence relation

(

U

,

s

)

∼

(

V

,

t

)

if there exists a

neighborhood

W

⊂

U

∩

V

of

p

such that

res

U

,

W

⁡

s

=

res

V

,

W

⁡

t

.

By

universal properties

of direct limit, a morphism

ϕ

:

F

⟶

G

of presheaves over

X

induces a morphism

ϕ

p

:

F

p

⟶

G

p

on each stalk

F

p

of

F

. Stalks are most useful in the

context

of sheaves, since they encapsulate all of the local data of the sheaf at the point

p

(recall that sheaves are basically defined as presheaves which have the property of being completely characterized by their local behavior). Indeed, in many of the standard examples of sheaves that take values in rings (such as the sheaf

𝒟

X

of smooth functions, or the sheaf

𝒪

X

of regular functions), the ring

F

p

is a local ring, and much of geometry is devoted to the study of sheaves whose stalks are local rings (so-called “

locally ringed spaces

”).

We mention here a few illustrations of how stalks accurately reflect the local behavior of a sheaf; all of these are drawn from

[

1

]

.

•

A morphism of sheaves

ϕ

:

F

⟶

G

over

X

is an

isomorphism

if and only if the

induced

morphism

ϕ

p

is an isomorphism on each stalk.

•

A

sequence

F

⟶

G

⟶

H

of morphisms of sheaves over

X

is an

exact sequence

at

G

if and only if the induced morphism

F

p

⟶

G

p

⟶

H

p

is exact at each stalk

G

p

.

•

The

sheafification

F

′

of a presheaf

F

has stalk equal to

F

p

at every point

p

.

References

1

Robin Hartshorne,

Algebraic Geometry

, Springer–Verlag New York Inc., 1977 (GTM

52

).

Title

stalk

Canonical name

Stalk

Date of creation

2013-03-22 12:37:15

Last modified on

2013-03-22 12:37:15

Owner

djao (24)

Last modified by

djao (24)

Numerical id

9

Author

djao (24)

Entry type

Definition

Classification

msc 54B40

Classification

msc 14F05

Classification

msc 18F20

Related topic

Sheaf

Related topic

LocalRing

Generated on Sat Feb 10 11:17:32 2018 by

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cofinite and cocountable topologies

cofinite and cocountable topologies

The

cofinite topology

on a set

X

is defined to be the

topology

𝒯

where

𝒯

=

{

A

⊆

X

∣

X

∖

A

⁢

is finite, or

⁢

A

=

∅

}

.

In other words, the

closed sets

in the cofinite topology are

X

and the finite subsets of

X

.

Analogously, the

cocountable topology

on

X

is defined to be the topology
in which the closed sets are

X

and the

countable

subsets of

X

.

The cofinite topology on

X

is the coarsest

T

1

topology (

http://planetmath.org/T1Space

) on

X

.

The cofinite topology on a

finite set

X

is the

discrete topology

.
Similarly, the cocountable topology on a

countable set

X

is the discrete topology.

A set

X

together with the cofinite topology forms a

compact

topological space

.

Title

cofinite and cocountable topologies

Canonical name

CofiniteAndCocountableTopologies

Date of creation

2013-03-22 13:03:30

Last modified on

2013-03-22 13:03:30

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

21

Author

yark (2760)

Entry type

Definition

Classification

msc 54B99

Related topic

FiniteComplementTopology

Defines

cofinite topology

Defines

cocountable topology

Defines

cofinite

Defines

cocountable

Generated on Sat Feb 10 11:17:36 2018 by

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cone

cone

Given a

topological space

X

, the

cone on

X

(sometimes denoted by

C

⁢

X

) is the

quotient space

X

×

[

0

,

1

]

/

X

×

{

0

}

.

Note that there is a natural inclusion

X

↪

C

⁢

X

which sends

x

to

(

x

,

1

)

.

If

(

X

,

x

0

)

is a

based topological space

, there is a similar

reduced cone

construction, given by

X

×

[

0

,

1

]

/

(

X

×

{

0

}

)

∪

(

{

x

0

}

×

[

0

,

1

]

)

.

With this definition, the natural inclusion

x

↦

(

x

,

1

)

becomes a based map, where we take

(

x

0

,

0

)

to be the basepoint of the reduced cone.

Title

cone

Canonical name

Cone

Date of creation

2013-03-22 13:25:20

Last modified on

2013-03-22 13:25:20

Owner

antonio (1116)

Last modified by

antonio (1116)

Numerical id

7

Author

antonio (1116)

Entry type

Definition

Classification

msc 54B99

Related topic

Suspension

Related topic

Join3

Defines

reduced cone

Generated on Sat Feb 10 11:17:38 2018 by

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initial topology

initial topology

Let

X

i

,

i

∈

I

be any family of

topological spaces

.
We say that a

topology

𝒯

on

X

is initial with respect to the family of

mappings

f

i

:

X

→

X

i

,

i

∈

I

, if

𝒯

is the coarsest topology on

X

which makes all

f

i

’s

continuous

.

The

initial topology

is characterized by the condition that a map

g

:

Y

→

X

is continuous if and only if every

f

i

∘

g

:

Y

→

X

i

is continuous.

Sets

𝒮

=

{

f

i

-

1

(

U

)

:

U

is open in

X

i

}

form a

subbase

for the initial topology, their finite intersections form a base.

E.g. the

product topology

is initial with respect to the projections (

http://planetmath.org/GeneralizedCartesianProduct

) and a

subspace topology

is initial with respect to the

embedding

.

The initial topology is sometimes called topology generated by a family of mappings

[

2

]

,

weak topology

[

4

]

or projective topology. (The weak topology is used mainly in

functional analysis

.)

From the viewpoint of

category theory

, the initial topology is an

initial source

. (Initial

structures

, which are a natural

generalization

of the initial topology, play an important rôle in topological categories and categorical topology.)

References

1

J. Adámek, H. Herrlich, and G. Strecker,

Abstract and

concrete categories

, Wiley,
New York, 1990.

2

R. Engelking,

General topology

, PWN, Warsaw, 1977.

3

M. Hušek,

Categorical topology

, Encyclopedia of General Topology (K. P. Hart,
J.-I. Nagata, and J. E. Vaughan, eds.), Elsevier, 2003, pp. 70–71.

4

S. Willard,

General topology

, Addison-Wesley, Massachussets, 1970.

5

Wikipedia’s entry on

http://en.wikipedia.org/wiki/Initial_topology

Initial topology

Title

initial topology

Canonical name

InitialTopology

Date of creation

2013-03-22 15:30:26

Last modified on

2013-03-22 15:30:26

Owner

kompik (10588)

Last modified by

kompik (10588)

Numerical id

11

Author

kompik (10588)

Entry type

Definition

Classification

msc 54B99

Related topic

producttopology

Related topic

subspacetopology

Related topic

ProductTopology

Related topic

IdentificationTopology

Related topic

Coarser

Generated on Sat Feb 10 11:17:40 2018 by

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join

join

Given two

topological spaces

X

and

Y

, their

join,

denoted by

X

⋆

Y

,

is defined to be the

quotient space

X

⋆

Y

:=

X

×

[

0

,

1

]

×

Y

/

∼

,

where the

equivalence relation

∼

is generated by

(

x

,

0

,

y

1

)

∼

(

x

,

0

,

y

2

)

for any

⁢

x

∈

X

,

y

1

,

y

2

∈

Y

,

and

(

x

1

,

1

,

y

)

∼

(

x

2

,

1

,

y

)

for any

⁢

y

∈

Y

,

x

1

,

x

2

∈

X

.

Intuitively,

X

⋆

Y

is formed by taking the

disjoint union

of the two spaces and attaching a

line segment

joining every point in

X

to every point in

Y

.

Some examples:

•

The join of a space

X

with a one-point space is called the

cone

of

X

.

•

The join of the spheres

S

n

and

S

m

is the sphere

S

n

+

m

+

1

.

Title

join

Canonical name

Join1

Date of creation

2013-03-22 13:25:40

Last modified on

2013-03-22 13:25:40

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 54B99

Related topic

Cone

Related topic

Suspension

Defines

join

Generated on Sat Feb 10 11:17:42 2018 by

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