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

TL;DR

This paper proves that for first countable paracompact sigma-spaces, the inductive and covering dimensions coincide, affirmatively answering a longstanding question about the equality of various dimension notions in such spaces.

Contribution

It establishes the equality of the $ ext{ind}$, $ ext{Ind}$, and $ ext{dim}$ dimensions for a class of first countable spaces with a countable network, resolving an open problem.

Findings

$ ext{ind} X = ext{Ind} X = ext{dim} X$ for the specified spaces

Positive resolution of Arkhangel'skii's question

Advances understanding of dimension theory in topological spaces

Abstract

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.