For Simplices
$ Def: Given a point P and an affine singular q-simplex σ = (P0, P1, ..., Pq) in an affine space, their join is the affine singular (q+1)-simplex

Pσ:= (P, P0, P1, ..., Pq) .

* Idea: The simplex obtained by "joining P with all the vertices of σ".
* Note: P doesn't have to be "outside σ", since these simplices are singular.
* For chains: If, instead of a simplex σ, we have a singular q-chain c, we define the join Pc by using linearity.
@ References: in Nash & Sen 83, p85.

PL Join of Subspaces X, Y of Rn
* Idea: The union of all line segments joining any xX to yY (provided no two such segments intersect).

For Topological Spaces X and Y
$ Def: The join of X and Y is the space X × I × Y / ~, where ~ is the equivalence relation

(x, 1, y) ~ (x', 1, y),      and      (x, 0, y) ~ (x, 0, y') .

* To visualize: Consider (X × Y) × I, and squeeze (X × Y) × {0} down to Y, and (X × Y) × {1} down to X.

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