Certain classes of weakly infinite-dimensional spaces and topological games

• Published in: Topology and its applications 2008-11, Vol.156 (1), p.61-69
• Main Authors: Fedorchuk, V.V., Osipov, E.V.
• Format: Article
• Language: English
• Subjects: C-space; Topological game; Transfinite dimension dim; Weakly infinite-dimensional space
• ISSN: 0166-8641; 1879-3207
• DOI: 10.1016/j.topol.2007.11.007

In [V.V. Fedorchuk, Questions on weakly infinite-dimensional spaces, in: E.M. Pearl (Ed.), Open Problems in Topology II, Elsevier, Amsterdam, 2007, pp. 637–645; V.V. Fedorchuk, Weakly infinite-dimensional spaces, Russian Math. Surveys 42 (2) (2007) 1–52] classes w– m– C of weakly infinite-dimensional spaces, 2 ⩽ m ⩽ ∞ , were introduced. We prove that all of them coincide with the class wid of all weakly infinite-dimensional spaces in the Alexandroff sense. We show also that transfinite dimensions dim w m , introduced in [V.V. Fedorchuk, Questions on weakly infinite-dimensional spaces, in: E.M. Pearl (Ed.), Open Problems in Topology II, Elsevier, Amsterdam, 2007, pp. 637–645; V.V. Fedorchuk, Weakly infinite-dimensional spaces, Russian Math. Surveys 42 (2) (2007) 1–52], coincide with dimension dim w 2 = dim , where dim is the transfinite dimension invented by Borst [P. Borst, Classification of weakly infinite-dimensional spaces. I. A transfinite extension of the covering dimension, Fund. Math. 130 (1) (1988) 1–25]. Some topological games which are related to countable-dimensional spaces, to C-spaces, and some other subclasses of weakly infinite-dimensional spaces are discussed.