Abstract
Assuming the continuum hypothesis we construct an example of a nonmetrizable compact set X with the following properties
(1) X n is hereditarily separable for all n ∈ ℕ
(2) X n \ Δ n is perfectly normal for every n ∈ ℕ, where Δ n is the generalized diagonal of X n, i.e., the set of points with at least two equal coordinates
(3) for every seminormal functor ℱ that preserves weights and the points of bijectivity the space ℱ k (X) is hereditarily normal, where k is the second smallest element of the power spectrum of the functor ℱ; in particular, X 2 and λ 3 X are hereditarily normal.
Our example of a space of this type strengthens the well-known example by Gruenhage of a nonmetrizable compact set whose square is hereditarily normal and hereditarily separable.
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Original Russian Text Copyright © 2008 Ivanov A. V. and Kashuba E. V.
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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 49, No. 4, pp. 813–824, July–August, 2008.
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Ivanov, A.V., Kashuba, E.V. Hereditary normality of a space of the form ℱ(X). Sib Math J 49, 650–659 (2008). https://doi.org/10.1007/s11202-008-0061-5
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DOI: https://doi.org/10.1007/s11202-008-0061-5