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Hurewicz properties, non distinguishing convergence properties and sequence selection properties. (English) Zbl 1059.03061

Several properties of a topological space \(X\) related to the behavior of open coverings of \(X\) or to the behavior of sequences of continuous real-valued functions on \(X\) are compared. Four types of properties are considered: (i) Hurewicz properties H*, H**, E*, E**, \(\text{E}_{\omega}^{\ast}\), \(\text{E}_{\omega}^{\ast\ast}\) [see L. Bukovský and J. Haleš, Topology Appl. 132, 71–79 (2003; Zbl 1056.54024)]. (ii) Properties related to non-distinguishing convergences of sequences of real functions: wQN, mQN, \(\Sigma\Sigma^{\ast}\), and \(\overline{\text{QN}}\) [see L. Bukovský, I. Recław and M. Repický, Topology Appl. 41, 25–40 (1991; Zbl 0768.54025)]. (iii) Rothberger covering properties \(\gamma\) and \(\text{C}''\) [see F. Rothberger, Fundam. Math. 30, 50–55 (1938; Zbl 0018.24701)]. (iv) Sequence selection properties SSP and MSS [see M. Scheepers, Proc. Am. Math. Soc. 125, 2789–2795 (1997; Zbl 0881.54038)]. The aim of the paper is showing that all those properties are closely connected (in fact, some of them are mutually equivalent).

MSC:

03E75 Applications of set theory
03E05 Other combinatorial set theory
54C30 Real-valued functions in general topology
26A99 Functions of one variable
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54C50 Topology of special sets defined by functions
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