Join (Spaces)

Let $X$ and $Y$ be Topological Spaces. Then their join is the factor space

$$X*Y =(X\times Y\times I)/\sim{},$$

where $\sim$ is the Equivalence Relation

$$(x,y,t) \sim (x',y',t') \Leftrightarrow \cases{ t=t'=0 {\rm... ...=x'\cr \hfill {\rm or}\hfill\cr t=t'=1 {\rm\ and\ } y=y.\cr}$$

See also Cone (Space), Suspension

References

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 6, 1976.