We show that for an open dense set of markets with a continuum of traders the number of equilibrium allocations [which by the celebrated theorem of Aumann (Econometrica, 1964, 32, 39-50) coincide with the core allocations for such markets] are finite. This presents a limiting case result that complements similar asymptotic theorems for cores of large economies proved by Bewley (Econometrica, 1973, 41, 425-454), and Dierker (Journal of Mathematical Economics, 1975, 2, 155-169). If we require that the measure on the space of agents be one with a finite number of atoms of equal weight, our reasoning recovers the classical results obtained by Debreu (Econometrica, 1970, 38, 387-392) for economies with a finite number of agents.