Functor M τ, Lipschitzian and uniformly continuous mappings

The paper is devoted to questions on lifting of the functor M τ: Tych → Tych to the categories of metric and uniform spaces. Similar problems were solved for the functor U τ of the unit ball of τ-additive measures. The main difference between the functor M τ and the functor U τ is that the space M τ(X) is compact only for X = 0. A more delicate distinction is expressed by Theorem 2 which implies that the functor M τ does not always preserve the uniform continuity of mappings of metric spaces (even in the case of compacta). Nevertheless, the problem of lifting the functor M τ to the category Unif turns to be solvable. © 2008 Allerton Press, Inc.