Normal functors and the metrizability of compact Hausdorff spaces (Q5960231)

For the normal functors in finite power \(\geq 3\) [\textit{E.~V.~Shchepin}, Russ. Math. Surv. 36, No. 3, 1-71 (1981); translation from Usp. Mat. Nauk 36, No. 3(219), 3-62 (1981; Zbl 0463.54009)] the following assertions are proved: a)~If for the bicompact \(X\) its subspace in hypersymmetric power \(\exp_3^* X\) is hereditarily normal, then the bicompact \(X\) is metrizable; b)~If for the bicompact \(X\) and the normal functor \(\mathcal F\) in power \(\geq 3\) the space \(\mathcal F^*(X)\) is hereditarily normal, the bicompact \(X\) is metrizable; c)~If for the normal functor \(\mathcal F\) in power \(\geq 3\) and the Hausdorff countably compact space \(X\) the space \(\mathcal F(X)\) is hereditarily normal, then \(X\) is a metrizable compact.