On the Dimension of Preimages of Certain Paracompact Spaces

It is proved that if X is a normal space which admits a closed fiberwise strongly zero-dimensional continuous map onto a stratifiable space Y in a certain class (an S-space), then IndX = dimX. This equality also holds if Y is a paracompact σ-space and ind Y = 0. It is shown that any closed network of a closed interval or the real line is an S-network. A simple proof of the Kateˇ tov–Morita inequality for paracompact σ-spaces (and, hence, for stratifiable spaces) is given.