On dimension of non-totally normal spaces (Q1174043)

In the class of normal spaces are considered the inductive dimensional invariants \(\text{Ind}_ 1\) and \(\text{ind}_ 1\), proposed by \textit{V. V. Filippov}, defined with the help of \(D\)-closed partitions, i.e. partitions, the complements of which are joins of a system of open sets, locally finite in these complements, of type \(F_ \sigma\) in the enveloping space. Briefly proved is the following sum theorem: If the sets \(F_ i\), \(i\in\mathbb{N}\), are \(D\)-closed in a normal space, then \(\text{Ind}_ 1\bigcup\{F_ i;\;i\in\mathbb{N}\}=\sup\{\text{Ind}_ 1 F_ i; i\in\mathbb{N}\}\). The following relations are stated: (1) \(\text{ind}_ 1\beta X\geq \text{Ind}_ 1 X\) for normal \(X\); (2) \(\text{ind}_ 1 X= \text{Ind}_ 1 X\) for completely paracompact \(X\); (3) \(\text{Ind}(X\times Y)\leq \text{Ind}_ 1 X+ \text{Ind}_ 1 Y\), if the product \(X\times Y\) is completely paracompact. (Translation from the Referativnyj Zhurnal Matematiki).