Join (topology)

In topology, a field of mathematics, the join of two topological spaces

A

{\displaystyle A}

and

B

{\displaystyle B}

, often denoted by

A

∗

B

{\displaystyle A\ast B}

or

A

⋆

B

{\displaystyle A\star B}

, is a topological space formed by taking the disjoint union of the two spaces, and attaching line segments joining every point in

A

{\displaystyle A}

to every point in

B

{\displaystyle B}

. The join of a space

A

{\displaystyle A}

with itself is denoted by

A

⋆

2

:=

A

⋆

A

{\displaystyle A^{\star 2}:=A\star A}

. The join is defined in slightly different ways in different contexts.

Geometric sets

If

A

{\displaystyle A}

and

B

{\displaystyle B}

are subsets of the Euclidean space

R

n

{\displaystyle \mathbb {R} ^{n}}

, then:

A

⋆

B

:=

{

t

⋅

a

+

(

1

−

t

)

⋅

b

|

a

∈

A

,

b

∈

B

,

t

∈

[

0

,

1

]

}

{\displaystyle A\star B\ :=\ \{t\cdot a+(1-t)\cdot b~|~a\in A,b\in B,t\in [0,1]\}}

,that is, the set of all line-segments between a point in

A

{\displaystyle A}

and a point in

B

{\displaystyle B}

.

Some authors restrict the definition to subsets that are joinable: any two different line-segments, connecting a point of A to a point of B, meet in at most a common endpoint (that is, they do not intersect in their interior). Every two subsets can be made "joinable". For example, if

A

{\displaystyle A}

is in

R

n

{\displaystyle \mathbb {R} ^{n}}

and

B

{\displaystyle B}

is in

R

m

{\displaystyle \mathbb {R} ^{m}}

, then

A

×

{

0

m

}

×

{

0

}

{\displaystyle A\times \{0^{m}\}\times \{0\}}

and

{

0

n

}

×

B

×

{

1

}

{\displaystyle \{0^{n}\}\times B\times \{1\}}

are joinable in

R

n

+

m

+

1

{\displaystyle \mathbb {R} ^{n+m+1}}

. The figure above shows an example for m=n=1, where

A

{\displaystyle A}

and

B

{\displaystyle B}

are line-segments.

Examples

The join of two simplices is a simplex: the join of an n-dimensional and an m-dimensional simplex is an (m+n+1)-dimensional simplex. Some special cases are:

The join of two disjoint points is an interval (m=n=0).