Certain classes of weakly infinite-dimensional spaces and topological games (Q958505)

In [\textit{V. V. Fedorchuk}, in E. Pearl (ed.), Open problems in topology II. Amsterdam: Elsevier (2007; Zbl 1158.54300), pp. 637--645; Russ. Math. Surv. 62, No.~2, 323--374 (2007); translation from Uspekhi Mat. Nauk 62, No.~2, 109--164 (2007; Zbl 1154.54022)], classes \(w\)-\(m\)-\(C\) of weakly infinite-dimensional spaces (\(m\) is an integer \(\geq 2\) or \(m=\infty\)) were introduced. The authors prove that all these notions coincide with the class \(w.i.d.\) of all weakly infinite-dimensional spaces in the Alexandroff sense. This question was asked in [op. cit.] and can be proven for normal spaces by an accessible argument. The notion of all \(S\)-\(w\)-\(m\)-\(C\) spaces coincides with the class of all weakly infinite-dimensional spaces in the Smirnov sense (\(S\)-\textit{w.i.d.} spaces). In [op. cit.] with the method from [\textit{P. Borst}, Fundam. Math. 130, No. 1, 1--25 (1988; Zbl 0661.54035)], transfinite dimension functions, \(\dim_{wm}\) where created to classify all \(S\)-\(w\)-\(m\)-\(C\) spaces. In this paper we see that all these functions also coincide with the function \(\dim_{w2} = \dim\) developed by Borst for \(S\)-\textit{w.i.d.} spaces. The main conclusion is that the class of \(w\)-\(m\)-\(C\) spaces soon after their introduction can be eliminated as a separate class of infinite-dimensional spaces. However, the same question for \(m\)-\(C\) spaces introduced in [op. cit.] remains still open. Using sequences of discrete families of closed subsets, \(w\)-\(C\) and \(S\)-\(w\)-\(C\) spaces are introduced. For collectionwise normal spaces these notions coincide with the class of \(w.i.d.\) and \(S\)-\textit{w.i.d.} spaces, respectively. Also the transfinite dimension function \(\dim_d\) developed for \(S\)-\(w\)-\(C\) spaces is identical with \(\dim\). The second part of the paper deals with topological games in connection with infinite dimensional notions as introduced in [\textit{I. Babinkostova}, Topol. Proc. 29, No.~1, 13--17 (2005; Zbl 1148.54327)]. Babinkostova proved for separable metric spaces that tp\((O,O)(X) \leq\omega\) iff \(X\) is countable dimensional. When we let \(O_m\) be the class of all open covers consisting of maximal \(m\) elements, then the authors prove that for a locally separable metric space \(\text{tp}_m(X) \equiv\text{tp}(O_m,O)(X) \leq\omega\) iff \(X\) is countable dimensional. Pol's type spaces are defined as compact metrizable spaces which are the union of a punctiform uncountable dimensional space and a countable dimensional space. Every Pol's type space is a C space. For each Pol's type space we have \(\text{tp}_m(X) = \omega+1\). Whether the converse is true is left as an open question. Another result is that for every \(m\)-\(C\) space \(X\) we have \(\text{tp}_m(X) < \omega_1\). The converse of this theorem is also left as a question.