On the dimension of preimages of certain paracompact spaces (Q722150)

The purpose of this paper is to investigate under which conditions the equality \(\dim X = \mathrm {Ind}\, X\) (the large inductive dimension of \(X\)) holds. A main result states: For each normal space \(X\), \(\dim X = \mathrm {Ind}\, X\) holds if \(X\) admits a closed strongly zero-dimensional continuous map \(f: X\to Y\) onto a stratifiable S-space \(Y\). Here, a map \(f: X\to Y\) is said to be strongly zero-dimensional if \(\dim f^{-1}(y) = 0\) for each \(y\in Y\). The equality also holds if \(Y\) is a paracompact \(\sigma\)-space with the small inductive dimension \(\mathrm {ind}\,Y = 0\). Furthermore, the author shows that any countable close network on a closed interval is an S-network, and gives a simple proof for the Katětov-Morita logarithmic inequality \(\dim (X\times Y) \leq \dim X + \dim Y\) for any finite-dimensional paracompact \(\sigma\)-spaces \(X\) and \(Y\).