Equality of Dimensions for Some Paracompact $\sigma$-Spaces

The equality of the dimensions $\operatorname{Ind}X$ and $\operatorname{dim}X$ of a first countable paracompact $\sigma$-space $X$ with a 1-continuous semimetric is proved. A partial positive answer to A. V. Arkhangel'skii's question about the equality of dimensions for first countable spaces with a countable network is given. As a consequence, the equality of the dimensions $\operatorname{Ind}X$ and $\operatorname{dim}X$ for Nagata spaces (that is, stratifiable first countable spaces) with a 1-continuous semimetric is obtained.