Lifting the functors $ U_\tau$ and $ U_R$ to the categories of bounded metric spaces and uniform spaces

Metric and uniform properties of the unit ball functors , , of measures with compact support, Radon measures, and -additive measures, respectively, are studied. It is proved that these functors can be lifted to the category BMetr of bounded metric spaces, BMetru of bounded metric spaces and uniformly continuous maps, and Unif of uniform spaces. Additionally, it is shown that the functor preserves the completeness property of metric spaces.

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