On metric spaces with given transfinite asymptotic dimensions (Q2140638)

Given a countable ordinal number \(\xi\), the authors construct a metric space \(X_\xi\) with \(\mathrm{transdim}(X_\xi) = \xi\) and \(\mathrm{coasym}(X_\xi) = \xi\). Here \(\mathrm{transdim}\) and \(\mathrm{coasym}\) denote transfinite asymptotic dimension in the sense of \textit{T. Radul} [Topology Appl. 157, No. 14, 2292--2296 (2010; Zbl 1198.54064)] and complementary-finite asymptotic dimension in the sense of \textit{Y. Wu} and \textit{J. Zhu} [ibid. 238, 90--101 (2018; Zbl 1387.54019)], respectively. This result implies that there exists an \(\infty\)-pseudometric space with asymptotic property C but without asymptotic property D.