On structural numbers of topological spaces
Vitalij Chatyrko, Alexandre Karassev
TL;DR
This paper introduces structural numbers Sn^{\mathcal{A}} for classes \mathcal{A} of hereditarily normal T_1-spaces, defined as the least cardinal of extensions whose intersection recovers the original topology, and connects them to the classical zero-dimensional numbers Z_0^{\mathrm{ind}} and Z_0^{\mathrm{dim}}. Central to the analysis is the Bing–Hanner construction, which produces extended topologies \tau(M) that preserve key dimensional and normality properties and express any topology as an intersection of zero-dimensional- or metrizable-structured extensions. The authors prove sharp bounds for Sn^{M_{\dim}} in metrizable spaces: for dim X = n \ge 0, 1 \le Sn^{M_{\dim}}(X) \le n+1, with a stronger lower bound when n \ge 1, and for countable-dimensional Y, 1 \le Sn^{M_{\dim}}(Y) \le \aleph_0; analogous results are obtained in non-metrizable contexts by decomposing X into unions of zero-dimensional subspaces. The work provides a framework linking extension of topologies with dimension theory, yielding concrete bounds and insights into how zero-dimensional components govern the behavior of structural numbers in both metrizable and non-metrizable settings.
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 $\mathrm{dim}\, Z = 0.$