On a Problem of Arkhangel'skii

TL;DR

This paper proves the equality of different topological dimensions for certain classes of spaces, confirming a conjecture about the coincidence of these dimensions in first countable spaces with a countable network.

Contribution

It establishes the equality of $ ext{Ind}$ and $ ext{dim}$ for first countable paracompact $\sigma$-spaces and confirms the conjecture that $ ext{ind}$, $ ext{Ind}$, and $ ext{dim}$ coincide for first countable spaces with a countable network.

Findings

Proved the coincidence of $ ext{Ind}$ and $ ext{dim}$ for specific spaces.

Confirmed that $ ext{Ind} X= ext{dim} X$ for Nagata spaces.

Provided a positive answer to Arkhangelskii's question on dimension coincidence.

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.