On structural numbers of topological spaces

TL;DR

This paper introduces and analyzes structural numbers related to topological spaces, extending previous concepts to subclasses of hereditarily normal spaces, and provides bounds for these numbers in metrizable and countable-dimensional spaces.

Contribution

It defines new structural numbers for subclasses of hereditarily normal spaces and establishes bounds for these numbers in specific classes of metrizable and countable-dimensional spaces.

Findings

For any metrizable space with dimension n, the structural number is between 1 and n+1.

For countable-dimensional metrizable spaces, the structural number is between 1 and aleph_0.

The paper extends the concept of structural numbers to subclasses of hereditarily normal T_1-spaces.

Abstract

Zero-dimensional structural numbers $Z_0^{\mathrm{ind}}$ and $Z_0^{\mathrm{dim}}$ w.r.t. dimensions $\mathrm{ind}$ and $\mathrm{dim}$ were introduced by Georgiou, Hattori, Megaritis, and Sereti. Somewhat similarly, we define structural numbers $\mathrm{Sn}^{A}$ for different subclasses $A$ of the class of hereditarily normal $T_1$-spaces. In particular, we show that: (a) for any metrizable space $X$ with $\dim X = n \geq 0$ we have $1 \leq \mathrm{Sn}^{M_{dim}}X \leq n+1$; (b) for any countable-dimensional metrizable space $Y$ we have $1 \leq \mathrm{Sn}^{M_{dim}}Y \leq \aleph_0$, where $ M_{dim}$ is the class of metrizable spaces $Z$ with $\mathrm{dim}\, Z = 0.$