Normality of partial products (Q1345685)

Necessary and sufficient conditions for the partial product [\textit{B. A. Pasynkov}, Tr. Mosk. Mat. O.-va 13, 136-245 (1965; Zbl 0147.413)] \(P_ k = \mathbb{P} (X, \{Z_ i\}^ k_ 1, \{O_ i\}^ k_ 1)\) to be normal (hereditarily normal, perfectly normal) are given. For example, it is proved the following Theorem 3: If a space \(X\) and the product \(S_{i_ 1 \ldots i_ m} = (O_{i_ 1} \cap \ldots \cap O_{i_ m}) \times Z_{i_ 1} \times \cdots \times Z_{i_ m}\) for all \(1 \leq i_ 1 < i_ 2 < \cdots < i_ m \leq k\) are hereditarily normal (perfectly normal) then the partial product \(\mathbb{P}_ k\) is also hereditarily normal (perfectly normal). If all spaces \(Z_ i\), \(1 \leq i \leq k\), are nonempty then the above mentioned conditions are also necessary for the partial product \(P_ k\) to be hereditarily normal (perfectly normal).