A seminormal functor F enjoys the Katětov property (K-property) if for every compact set X the hereditary normality of F (X) implies the metrizability of X. We prove that every seminormal functor of finite degree n > 3 enjoys the K-property. On assuming the continuum hypothesis (CH) we characterize the weight preserving seminormal functors with the K-property. We also prove that the nonmetrizable compact set constructed in [1] on assuming CH is a universal counterexample for the K-property in the class of weight preserving seminormal functors.In 1981 Shchepin defined [2] normal functors, thereby laying foundation for the theory of covariant functors in the category Comp of compact sets and continuous mappings. A natural weakening of the normality condition yields the concept of a seminormal functor (see [3]) which includes the well-known topological constructions like raising to a power, the exponential, the superextension λ, the space P of probability measures, and others. It makes possible the functorial interpretations of a series of classical results of general topology. This includes the theorem of Katětov [4] stating that every compact set whose cube is hereditarily normal is metrizable. Say that a seminormal functor F enjoys the Katětov property (K-property) if for every compact set X the hereditary normality of F (X) implies the metrizability of X. By the Katětov theorem the functor of raising to the third power enjoys the K-property.Several articles are devoted to the question of extending the Katětov theorem to various classes of functors. Fedorchuk proved [5] that all normal functors of degree ≥ 3 enjoy the K-property; Zhuraev established [6] the K-property for λ 4 ; the author proved (see [7]) the K-property for the functors of the form F n , where F is a seminormal functor satisfying some combinatorial condition ( * ) with the degree spectrum sp F = {1, k, n, . . . }. Generalizations of the Katětov theorem in the direction of weakening the hereditary normality condition on F (X) were obtained by Kombarov [8] for normal functors and Kashuba [7] for the seminormal functors satisfying ( * ).It is known that the question of the K-property for the squaring functor is not solvable in ZF C (the famous Katětov's problem): some counterexamples were constructed [9] by Nyikos (on assuming the Martin axiom and the negation of CH) and Gruenhage (on assuming CH); Larson and Todorčević proved [10] the consistency of the positive solution with ZF C. A nonmetrizable compact set X 0 is constructed in [1] on assuming CH such that for every functor F preserving weight and bijectivity points with the degree spectrum sp F = {1, k, . . . } the space F k (X 0 ) is hereditarily normal. In particular, the functor λ 3 lacks the K-property on assuming CH.In this article we prove (Theorem 3) that every seminormal functor F of finite degree n > 3 enjoys the K-property. 1) Therefore, there is the maximal possible degree of a seminormal functor lacking the K-property. On assuming CH we completely describe (Theorem 4) the weight pre...
