Smash product

Definition

Given two topological spaces with basepoint, $(X,x_0)$ and $(Y,y_0)$, their smash product is defined as the quotient of $X\times Y$ by the following equivalence relation:

$(x,y_0)\simeq (x_0,y)$

In other words, we collapse both the copy of $X$ and the copy of $Y$, through $(x_0,y_0)$, to a single point (the union of the $X$-copy and $Y$-copy is isomorphic to the wedge sum of the spaces, hence the smash product can be viewed as the quotient of the product by collapse of the wedge sum to a point).

Particular cases

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