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On a Problem of Arkhangel'skii

I. M. Leibo

Abstract

The coincidence of the $\Ind$ and $\dim$ dimensions for first countable paracompact $σ$-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.

On a Problem of Arkhangel'skii

Abstract

The coincidence of the and dimensions for first countable paracompact -spaces is proved. As a corollary, the equality for every Nagata (that is, first countable stratifiable) space is obtained. This gives a positive answer to A.~V.~Ar\-khan\-gel'\-skii's question of whether the dimensions , , and coincide for first countable spaces with a countable network.

Paper Structure

This paper contains 3 sections, 16 theorems, 20 equations.

Table of Contents

  1. Basic Definitions, Known Facts, and Auxiliary Results
  2. Main Theorems
  3. Open Problems

Key Result

Theorem 1.5

If $X$ is an f-space, then $\operatorname{Ind} X=\dim X$.

Theorems & Definitions (34)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.4: 5
  • Theorem 1.5: 5, 15
  • Theorem 1.6: 5, 15
  • Theorem 1.7: 5, 15
  • Definition 1.8
  • Theorem 1.9: 5
  • Theorem 1.10: 9
  • Remark 1.12
  • ...and 24 more