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Abstract We study global properties of the equilibrium set of economies with a continuous consumption space. This framework is important in intertemporal allocation problems (continuous time), financial markets with uncertainty (continuous states of nature) and models of commodity differentiation. We show that the equilibrium set is contractible which implies that (i) there is a continuous economic policy linking any two equilibrium states, and (ii) any two such economic policies can be continuously deformed one into the other. We also give three equivalent formulations of the problem of global uniqueness of equilibria in terms of the projection map from the equilibrium set to the space of parameters. We finally study the local and global effects that the existence of critical economies has on the equilibrium set.