Reduced suspension

Definition
Given a topological space with basepoint $$(X,x_0)$$, the reduced suspension of $$X$$ is defined in the following equivalent ways:


 * It is the smash product of $$(X,x_0)$$ with $$(S^1,p)$$ (we could choose $$p$$ to be any point in $$S^1$$ (it does not matter since $$S^1$$ is a homogeneous space).
 * It is the quotient of the suspension of $$X$$ by the identification:

$$(x_0,t) \sim (x_0,t')$$

The reduced suspension of $$X$$ is denoted by $$\Sigma X$$.

Facts
The reduced suspension commutes with joins. In other words:

$$\Sigma(X * Y) = \Sigma X * \Sigma Y$$