In this study, we explore the connection between the dimension of a pure exchange smooth economy and the topology of the set of the critical equilibria. In particular, using the dual framework developed by Balasko, we show that the set of regular equilibria is path connected if two consumers (goods) and at least four goods (consumers) exist, or if the sum of the number of goods and consumers is even. This result depends mainly on the fact that a real projective space of dimension n, ℝℙn, which is embedded in ℝℙn+1, does not disconnect ℝℙn+1 unless n=1.