On a Problem of Arkhangel'skii

The coincidence of the $\text{Ind}$ and $\dim$ dimensions for first countable paracompact $\sigma$-spaces is proved. As a corollary, the equality $\text{Ind}X=\dim X$ for every Nagata (that is, first countable stratifiable) space $X$ is obtained. This gives a positive answer to A.~V.~Ar-khan-gel’-skii’s question of whether the dimensions $\text{ind}$, $\text{Ind}$, and $\dim$ coincide for first countable spaces with a countable network.