Join of topological spaces

Definition
Given two topological spaces $$X$$ and $$Y$$, the join of $$X$$ and $$Y$$, denoted $$X * Y$$, is defined as follows: it is the quotient of the space $$X \times Y \times I$$ under the identifications:

$$(x,y_1,0) \sim (x,y_2,0) \forall x \in X, y_1,y_2 \in Y$$

and

$$(x_1,y,1) \sim (x_2,y,1) \forall x_1,x_2 \in X, y \in Y$$

Pictorially, we can think of this as the space of all line segments joining points in $$X$$ and $$Y$$, with two line segments meeting only at common endpoints.

Cone space
The cone space of a topological space $$X$$ can be viewed as the join of $$X$$ with a one-point space.

Suspension
The suspension of a topological space $$X$$ can be viewed as the join of $$X$$ with a two-point space.

Simplex
The $$n$$-simplex can be viewed, at least topologically, as the join of $$n$$ one-point spaces.

Operation properties
There is a canonical isomorphism between $$X * Y$$ and $$Y * X$$, sending $$(x,y,t)$$ to (y,x,1-t).

There is a canonical isomorphism between $$(X * Y) * Z$$ and $$X * (Y * Z)$$.