Coincidence of the Dimensions of First Countable Spaces with a Countable Network

The coincidence of the $\Ind$ and $\dim$ dimensions for first countable paracompact $\sigma$-spaces is proved. This gives a positive answer to A. V. Arkhangel'skii's question of whether the dimensions $\ind X$, $\Ind X$, and $\dim X$ are equal for first countable spaces with a countable network.