Understanding Smash Product Suppose I have pointed topological spaces $(X,x_0)$ and $(Y,y_0)$, then I would like to understand $(X,x_0) \land (Y,y_0)$. Since the quotient space can be defined by a partition, I will try to define it in that way.
From what I read, I think that the smash product is the quotient space of $X \times Y $ where the partition is given as follows. if $$ A=X \times \{y_0\} \cup \{x_0\} \times Y  $$ then the partition consists of the class $A$ and rest are singleton sets of the form $X \times \{y\}$  or  $\{x\} \times Y$ where $x \neq x_0$ and $y \neq y_0$. Am I right?
If I am right, can anyone explain why are we using such spaces? Are these really helpful?
 A: Regarding intuition:
I think its easier to think of the quotient as the wedge product, because at least then its easier to visualize what we are quoteinting out by. For example, with $S^1 \wedge S^1$ we are taking the product $S^1 \times S^1 $ which is the torus, and quotienting out by the wedge of two circles. You should draw a torus, with two loops on it, each representing the generators of H_1, and then work out what you get if yo collapse the two loops to a single point.
And since examples are super helpful for intuition, you should try to prove that smashing X by $S^1$ is the same as suspension of $X$.
Regarding why we care:
In most cases, I don't think we visualize the smash product, because its usually really hard to picture. The smash product is important though because it basically acts like tensor products in algebra. Namely, with the tensor product, we have the adjunction $Hom(X,Hom(Y,Z)) \cong  Hom(X \otimes Y, Z)$. We have the same thing with smash product in the category of pointed spaces: $Maps(X, Maps(Y,Z)) \cong Maps(X \wedge Y, Z)$.
From a more advanced and rather hand-wavy viewpoint, this product extends to spectra, which are basically sequences of spaces that stabilize after applying suspension enough. The example with it being suspension in the case that one of the spaces is $S^1$ indicates that it sort of "plays nicely" with spectra.
