Initial and universal metric spaces

Abstract:

Local capacity is introduced (the usual notion of the capacity is global). It is proven (see Theorem 1) that some classes of metric spaces, naturally defined in terms of local capacity, contain a space which can be mapped onto any other member of its class without any stretching. Such a theorem would fail if local capacity is replaced by the usual notion of (global) capacity. Using Theorem 1 and simple properties of Met $(X,Y)$ (see §2) it follows immediately that for every class of compact metric spaces with uniformly bounded diameters and capacities there exists a compact space which contains an isometric image of any space from the class (in general this universal space cannot be found within the class).