In the mathematical statistics book, the support of $f(x_1,x_2)$ is the Cartesian product $S_1 \times S_2.$ This is a generalization of the slightly more elementary 'rectangular' space in the actuarial review book, where the two supports are intervals (the usual case in practice).
In either case, the intent is to exclude an example such as the following:
Let the joint distribution be uniform over the triangle with vertices at
(0,0), (0,1), and (1,0). It is easy to show that the two random variables
with this joint PDF are not independent. The two marginals are BETA(1,2), each with the unit
interval as support.
However, the product of the two beta marginal PDFs is not
the same as the original distribution. This product produces a joint PDF
with the entire unit square as its support. And of course its two associated random variables are independent.