On structural numbers of topological spaces

Abstract

Zero-dimensional structural numbers $Z_0^{\mathrm{ind}}$ and $Z_0^{\dim }$ w.r.t. dimensions $\mathrm{ind}$ and $\dim$ 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 $\dim X=n\ge 0$ we have $1\le {\mathrm{Sn}}^{M_{dim}}X\le n+1$; (b) for any countable-dimensional metrizable space $Y$ we have $1\le {\mathrm{Sn}}^{M_{dim}}Y\le {\aleph }_0$, where $M_{dim}$ is the class of metrizable spaces $Z$ with $\dim Z=0.$