On a dimension-like functions and its behavior under continuous mappings of special spaces (Q2703093)

The author uses \textit{L. G. Zambakhidze}'s [Tr. Tbilis. Mat. Inst. Razmadze 56, 52-98 (1977; Zbl 0396.54033)] integer valued function \(I(X)\) on Tikhonov spaces \(X\) that describes a compactness degree of remainders (\(I(X)=0\) iff \(X\) is nonempty compact, \(I(X)=1\) iff \(X\) is noncompact locally compact, \dots). Generalizing Parkhomenko's result (every locally compact space has a coarser compact topology) the author proves that every \(X\) with odd \(I(X)\) is finer than some \(Y\) with \(I(Y)=I(X)-1\) and \(w(X)\geq w(Y)\). Examples are given showing that \(I(Y)\) cannot be decreased and that the result is not true for even \(I(X)\); however, such examples do not exist if \(X\) is metrizable, separable and zerodimensional. At the end, a modification of the original definition of \(I(f)\) for continuous mappings \(f:X\to Y\) is given such that the inequality \(I(X)\leq I(f)+I(Y)\) always holds.