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). The join of a point and an interval is a triangle (m=0, n=1). The join of two line segments is homeomorphic to a solid tetrahedron or disphenoid, illustrated in the figure above right (m=n=1). The join of a point and an (n-1)-dimensional simplex is an n-dimensional simplex. The join of a point and a polygon (or any polytope) is a pyramid, like the join of a point and square is a square pyramid. The join of a point and a cube is a cubic pyramid. The join of a point and a circle is a cone, and the join of a point and a sphere is a hypercone.
Topological spaces If A {\displaystyle A} and B {\displaystyle B} are any topological spaces, then:
A ⋆ B := A ⊔ p 0 ( A × B × [ 0 , 1 ] ) ⊔ p 1 B , {\displaystyle A\star B\ :=\ A\sqcup _{p_{0}}(A\times B\times [0,1])\sqcup _{p_{1}}B,}
where the cylinder A × B × [ 0 , 1 ] {\displaystyle A\times B\times [0,1]} is attached to the original spaces A {\displaystyle A} and B {\displaystyle B} along the natural projections of the faces of the cylinder:
A × B × { 0 } → p 0 A , {\displaystyle {A\times B\times \{0\}}\xrightarrow {p_{0}} A,}
A × B × { 1 } → p 1 B . {\displaystyle {A\times B\times \{1\}}\xrightarrow {p_{1}} B.}
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