The Concept of Separable Connectedness: Applications to General Utility Theory

We say that a topological set X is separably connected if for any two points x, y ∈ X there exists a connected and separable subset C(x, y) ⊆ X to which both x and y belong. This concept generalizes path-connectedness. With this concept we have improved some results on general utility theory: For instance, in 1987 Monteiro gave conditions (dealing with real-valued, continuous, order-preserving functions) on path-connected spaces in order to get continuous utility representations of continuous total preorders defined on the set. We have recently proved (in an article by Candeal, Herves and Indurain, published in the Journal of Mathematical Economics, 1998) that Monteiro’s results also work for the more general case of separably connected spaces. Then we study the particular situation of separable connectedness on spaces endowed with some extra structure, e.g. metric spaces.