On a Problem of Arkhangel'skii

Abstract

The coincidence of the $\Ind$ and $\dim$ dimensions for first countable paracompact $\sigma$-spaces is proved. As a corollary, the equality $\Ind X= \dim X$ 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 $\ind$, $\Ind$, and $\dim$ coincide for first countable spaces with a countable network.