We shall present an alternative proof of G. Debreu's theorem on the finiteness of the set of equilibria [1], and some related new results. The set of economies with finitely many equilibria is dense with respect to a very weak topology on the set of exchange economies. The equilibrium correspondence and the number of equilibria are continuous with respect to a strong topology on the set of regular economies where "regular" is defined as in [1]. Perhaps the methods used in this paper are more interesting than the results. The key concept we use is that of transversality. The essential regularity property of regular economies is that their excess demand functions are transversal to zero.