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