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join (Definition)

Given two topological spaces $ X$ and $ Y$, their join, denoted by $ X\star Y,$ is defined to be the quotient space

$\displaystyle X\star Y := X\times [0,1]\times Y/\sim, $
where the equivalence relation $ \sim$ is generated by
$\displaystyle (x,0,y_1)$ $\displaystyle \sim (x,0,y_2)$ for any$\displaystyle \, x\in X,\, y_1,y_2\in Y,\,$   and  
$\displaystyle (x_1,1,y)$ $\displaystyle \sim (x_2,1,y)$ for any$\displaystyle \, y\in Y,\, x_1,x_2\in X.$  

Intuitively, $ X\star Y$ is formed by taking the disjoint union of the two spaces and attaching a line segment joining every point in $ X$ to every point in $ Y.$

Some examples:

  • The join of a space $ X$ with a one-point space is called the cone of $ X$.
  • The join of the spheres $ S^n$ and $ S^m$ is the sphere $ S^{n+m+1}$.



"join" is owned by mathcam. [ full author list (2) | owner history (1) ]
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See Also: cone, suspension

Also defines:  join
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Cross-references: spheres, cone, point, line segment, disjoint union, generated by, equivalence relation, quotient space, topological spaces
There are 5 references to this entry.

This is version 4 of join, born on 2003-02-06, modified 2004-09-13.
Object id is 3985, canonical name is Join3.
Accessed 4050 times total.

Classification:
AMS MSC: 54B99 (General topology :: Basic constructions :: Miscellaneous)
Pending Errata and Addenda
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