ULTRAMETRICS AND INFINITE-DIMENSIONAL WHITEHEAD THEOREMS IN SHAPE-THEORY

We apply a Cantor completion process to construct a complete, non-Arch imedean metric on the set of shape morphisms between pointed compacta. In the case of shape groups we obtain a canonical norm producing a co mplete, both left and right invariant ultrametric. On the other hand, we give a new characterization of movability and we use these spaces o f shape morphisms and uniformly continuous maps between them, to prove an infinite-dimensional theorem from which we can show, in a short an d elementary way, some known Whitehead type theorems in shape theory.