Why does not being rectangular support implies dependence? In my book it states that "Whenever the support S is not “rectangular,” the random variables must be dependent, because
S cannot then equal the product set $\{(x, y) : x ∈ S_X , y ∈ S_Y \}$. That is, if we observe that
the support S of X and Y is not a product set, then X and Y must be dependent."
I know being independent means $f_{XY}(x,y)=f_X(x)f_Y(y)$ or $f_X(X|Y)=f_X(X|Y)=f_X(x)$ and same for $f_Y(y|X)$.
I don't see the relationship here. Why not having rectangular support mean it must be dependent?
 A: To illustrate the concept with an example: suppose $(X, Y)$ is uniformly distributed on the triangle $\{(x, y) : 0 \leq x \leq y \leq 1\}$. The marginal support of either $X$ or $Y$ is $[0, 1]$, meaning that individually, they can assume any value on the unit interval. However, if you know that $X = 0.6$, suddenly having $Y < 0.3$ becomes impossible, even though it should have been possible in a vacuum. If conditioning on a zero-probability event bothers you, note that the same thing would be true with just knowing that $0.6 \leq X \leq 1$, for instance.
This is the "problem" with non-rectangular supports, which you can see more clearly by looking at marginal distributions. If the marginal supports for $X$ and $Y$ are $[a, b]$ and $[c, d]$, and the joint support of $(X, Y)$ is not $[a, b] \times [c, d]$, then you have some impossible combination of values for the two variables. This is precisely how knowledge of one variable can communicate information about the other, which is exactly what it means to have dependent variables.
