On structural numbers of topological spaces

Abstract Zero-dimensional structural numbers <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mrow> <m:mi class="MJX-tex-caligraphic" mathvariant="script">Z</m:mi> </m:mrow> <m:mn>0</m:mn> <m:mrow> <m:mtext>ind </m:mtext> </m:mrow> </m:msubsup> <m:mtext> and </m:mtext> <m:msubsup> <m:mrow> <m:mi class="MJX-tex-caligraphic" mathvariant="script">Z</m:mi> </m:mrow> <m:mn>0</m:mn> <m:mrow> <m:mtext>dim </m:mtext> </m:mrow> </m:msubsup> </m:math> $\mathcal{Z}_0^{\text {ind }} \text { and } \mathcal{Z}_0^{\text {dim }}$ w.r.t. dimensions ind and dim were introduced by Georgiou, Hattori, Megaritis, and Sereti. Somewhat similarly, we define structural numbers Sn 𝒜 for different subclasses 𝒜 of the class of hereditarily normal T 1 -spaces. In particular, if ℳ dim denotes the class of metrizable spaces Z with dim Z = 0 ; we show that: for any metrizable space X with dim X = n ≥ 0, we have <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>1</m:mn> <m:mo>≤</m:mo> <m:mrow> <m:msup> <m:mo form="prefix">Sn</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi class="MJX-tex-caligraphic" mathvariant="script">M</m:mi> </m:mrow> <m:mrow> <m:mo form="prefix">dim</m:mo> </m:mrow> </m:msub> </m:mrow> </m:msup> </m:mrow> <m:mi>X</m:mi> <m:mo>≤</m:mo> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> <m:mo>;</m:mo> </m:math> $1 \leq \operatorname{Sn}^{\mathcal{M}_{\operatorname{dim}}} X \leq n+1 ;$ for any countable-dimensional metrizable space Y , we have <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>1</m:mn> <m:mo>≤</m:mo> <m:mrow> <m:msup> <m:mo form="prefix">Sn</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi class="MJX-tex-caligraphic" mathvariant="script">M</m:mi> </m:mrow> <m:mrow> <m:mo form="prefix">dim</m:mo> </m:mrow> </m:msub> </m:mrow> </m:msup> </m:mrow> <m:mi>Y</m:mi> <m:mo>≤</m:mo> <m:msub> <m:mi mathvariant="normal">ℵ</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>.</m:mo> </m:math> $1 \leq \operatorname{Sn}^{\mathcal{M}_{\operatorname{dim}}} Y \leq \aleph_0 .$