Abstract
In this paper the inner and categorical characterizations of perfect mappings have been obtained in pull-back terms. Frolik in 1974 devoted a small number of works to the study of this category. Although ZUnif is strictly intermediate between the fairly studied categories Unif and Tych (Tikhonov spaces and continuous mappings). Solving various problems in the ZUnif category naturally entails generalizing similar problems in the Unif category and strengthening similar problems in the Tych category. Particular interest in studying and constructing the ZUnif category in theory arose after the construction in the work of a similar compactification of a uniform space, where the extendable mappings turned out to be morphisms of uniform spaces.
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