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 $\operatorname{Ind}{X}=\operatorname{dim}{X}$. This equality also holds if ${Y}$ is a paracompact $\sigma$-space and $\operatorname{ind}{Y}=0$. It is shown that any closed network of a closed interval or the real line is an S-network. A simple proof of the Katětov–Morita inequality for paracompact $\sigma$-spaces (and, hence, for stratifiable spaces) is given.