On structural numbers of topological 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 $\mathrm{dim}\, Z = 0.$