We say that a topological set X is separably connected if for any
two points $x,\; y\;\epsilon\; X$ there exists a connected and separable
subset $C\left(x,y\right)\subseteq X$ 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 defi{}ned
on the set. We have recently proved (in an article by Candeal, Hervès
and Indurain, published in the Journal of Mathematical Economics,
1998) that Monteiro\textquoteright{}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.
