Abstract
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 = O. 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.
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