  The standard method to check for the independence of two real-valued random
variables -- demonstrating that the bivariate joint distribution factors into
the product of its marginals -- is both necessary and sufficient. Here we
present a simple necessary condition based on the support sets of the random
variables, which -- if not satisfied -- avoids the need to extract the
marginals from the joint in demonstrating dependence. We review, in an
accessible manner, the measure-theoretic, topological, and probabilistic
details necessary to establish the background for the old and new ideas
presented here. We prove our result in both the discrete case (where the basic
ideas emerge in a simple setting), the continuous case (where serious
complications emerge), and for general real-valued random variables, and we
illustrate the use of our condition in three simple examples.
