Join (topology)

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

where I is the interval [0, 1] and R is the equivalence relation generated by

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

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