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).


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
Full Text: EuDML