Join (topology)

In topology, a field of mathematics, the join of two topological spaces ${\displaystyle A}$ and ${\displaystyle B}$, often denoted by ${\displaystyle A\ast B}$ or ${\displaystyle A\star B}$, is a topological space defined as

${\displaystyle A\star B\$ :=\ A\sqcup _{p_{0}}(A\times B\times [0,1])\sqcup _{p_{1}}B,}

Geometric join of two line segments. The original spaces are shown in green and blue. The join is a three-dimensional solid in gray.

where the cylinder ${\displaystyle A\times B\times [0,1]}$ is attached to the original spaces ${\displaystyle A}$ and ${\displaystyle B}$ along the natural projections of the faces of the cylinder:

${\displaystyle {A\times B\times \{0\}}\xrightarrow {p_{0}} A,}$

${\displaystyle {A\times B\times \{1\}}\xrightarrow {p_{1}} B.}$

Intuitively, ${\displaystyle A\star B}$ is formed by taking the disjoint union of the two spaces and attaching line segments joining every point in ${\displaystyle A}$ to every point in ${\displaystyle B}$.

Note that usually it is implicitly assumed that ${\displaystyle A}$ and ${\displaystyle B}$ are non-empty, in which case the definition is often phrased a bit differently: instead of attaching the faces of the cylinder ${\displaystyle A\times B\times [0,1]}$ to the spaces ${\displaystyle A}$ and ${\displaystyle B}$, these faces are simply collapsed in a way suggested by the attachment projections ${\displaystyle p_{1},p_{2}}$:

${\displaystyle A\star B\$ :=\ (A\times B\times [0,1])/R,}

where ${\displaystyle R}$ is the equivalence relation generated by

${\displaystyle (a,b_{1},0)\sim (a,b_{2},0)\quad {\mbox{for all }}a\in A{\mbox{ and }}b_{1},b_{2}\in B,}$

${\displaystyle (a_{1},b,1)\sim (a_{2},b,1)\quad {\mbox{for all }}a_{1},a_{2}\in A{\mbox{ and }}b\in B.}$

At the endpoints, this collapses ${\displaystyle A\times B\times \{0\}}$ to ${\displaystyle A}$ and ${\displaystyle A\times B\times \{1\}}$ to ${\displaystyle B}$.