On universal spaces and absorbing sets related to a transfinite extension of covering dimension
In the realm of transfinite dimensions there are various results on universal spaces. For most kinds of transfinite dimensions such spaces do not exist. For the small transfinite dimension, trind, R. Pol has shown that for every countable ordinal number \(\alpha\) there exists a universal space for separable metrizable spaces \(X\) with trind \(X \leq \alpha\). W. Olszewski has shown that for the large transfinite dimension trInd and each countable limit ordinal number \(\lambda\) there is no universal space with trInd \(X \leq \lambda\). For the transfinite extension of the covering dimension, \(\dim_w\), introduced by P. Borst in the realm of weakly infinite dimensional spaces, the author and M. Zarichnyi have proved that for every countable ordinal limit ordinal there is no universal space with \(\dim_w X \leq \alpha\). In this paper the author proves the same result for the transfinite dimension \(\dim_C\). Dim\(_C\) is also a transfinite extension of the covering dimension introduced by P. Borst is the realm of \(C\)-spaces. The construction of the compacta \(D^{\sigma}_\lambda\) which is used to force a contradictionary situation was developed by \textit{W. Olszewski} [Fundam. Math. 144, No. 3, 243--258 (1994; Zbl 0812.54041)]. The proof uses some useful lemmas on dim\(_C\). Like the discrete sum theorem and the proof that \(\dim_C S_\alpha = \alpha\) for the Smirnov space \(S_\alpha\). It is proven that for each countable limit ordinal number \(\lambda\) there exists no universal space for spaces \(X\) with \(\dim_{C} \leq \lambda\). Even the stronger result is proven that there exists no universal space for compact metric spaces \(X\) with \(\dim_C \leq \lambda\). Not mentioned in the paper is the related result of \textit{T. Radul} [Topology Appl. 83, No. 2, 127--133 (1998; Zbl 0930.54027)] in which he proves that for \(C\)-compact metric spaces with \(\dim_C \leq \lambda\) there is an absorbing space which has \(\dim_C \leq \alpha\). In addition as a corollary a result on absorbing spaces is proven in which it is stated that for each countable limit ordinal number \(\xi\) there is no absorbing set for \(\{X \in C : \dim_C X \leq \xi\}\) for each collection \(C \in \{\mathbf M_\alpha : \alpha < \overline{\omega}_1\}\cup \{{\mathbf A}_\alpha : \alpha < \overline{\omega}_1\}\). We denote by \({\mathbf M}_\alpha\) (respectively \({\mathbf A}_\alpha\)) the absolute multiplicative (respectively, additive) Borel class of order \(\alpha\) [see \textit{M. Bestvina} and \textit{J. Mogilski}, Mich. Math. J. 33, 291--313 (1986; Zbl 0629.54011)].
scientific article; zbMATH DE number 5151665