On inductive dimensions (Q2266250)

The author defines inductive dimension \(\sigma\) Ind as \(\sigma\) Ind X\(=- 1\) iff \(X=\emptyset\). Assume that the class of spaces X with \(\sigma\) Ind X\(<\alpha\) is already defined. Then \(\sigma Ind X\leq \alpha\) if X can be decomposed as countable sum of closed subsets \(F_ i\), \(i=1,2,..\). with the property: for each pair of disjoint closed subsets A and B of X there exist partitions \(C_ i\) in \(F_ i\) between \(A\cap F_ i\) and \(B\cap F_ i\) with \(\sigma Ind C_ i<\alpha\), \(i=1,2,... \). Analogously we get a definition of \(\sigma_ fInd\) if the word ''countable'' above is changed with the word ''finite''. If the set B changes with a point then we get the definitions of \(\sigma\) ind and \(\sigma_ find\) respectively. The author establishes some facts concerning the above mentioned notions. In particular some relations are obtained between the inductive dimensions and the classical ''dim'' and ''Ind'' and also some factorization theorems and assertions concerning some compact extensions and universal spaces. The author investigates the problem for the verification of the equality \(\sigma\) Ind X\(=\sigma Ind \beta X\) of any normal space X.