Kočinac, Ljubiša D. R.; Konca, Şükran Set-Menger and related properties. (English) Zbl 1437.54020 Topology Appl. 275, Article ID 106996, 9 p. (2020). Let \(\mathcal P\) be a collection of subsets of a space \(X\) and say that \(X\) is \(\mathcal P\)-Menger (resp. weakly \(\mathcal P\)-Menger, almost \(\mathcal P\)-Menger) if for each \(A\in\mathcal P\) and each sequence \(\langle\mathcal U_n\rangle\) of families of open subsets such that \(\overline{A}\subset\cup\mathcal U_n\) for each \(n\) there is a sequence \(\langle\mathcal V_n\rangle\) such that each \(\mathcal V_n\) is a finite subset of \(\mathcal U_n\) and \(A\subset\cup_{n\in\omega}\cup\mathcal V_n\) (resp. \(A\subset\overline{\cup_{n\in\omega}\cup\mathcal V_n}\), \(A\subset\cup_{n\in\omega}\overline{\cup\mathcal V_n}\)); if each \(\mathcal V_n\) can be taken as a singleton then the word “Menger” is replaced by “Rothberger.” When \(\mathcal P=\{X\}\) then \(\mathcal P\) may be dropped and when \(\mathcal P\) is the power set of \(X\) then \(X\) is said to be set-Menger, etc. Hurewicz and Gerlits-Nagy versions are also given but the emphasis is on set-Menger and set-Rothberger, especially their behaviour under subspaces and (pre-)images. Topological games are described and it is shown that if \(X\) is Lindelöf then \(X\) is (weakly, almost) set-Menger if and only if the first player does not have a winning strategy in the respective game. Reviewer: David B. Gauld (Auckland) Cited in 1 ReviewCited in 12 Documents MSC: 54D20 Noncompact covering properties (paracompact, Lindelöf, etc.) 54B05 Subspaces in general topology 54C10 Special maps on topological spaces (open, closed, perfect, etc.) 54G10 \(P\)-spaces 91A44 Games involving topology, set theory, or logic Keywords:selection principles; game theory; Menger; set-Menger; quasi-Menger; P-space PDFBibTeX XMLCite \textit{L. D. R. Kočinac} and \textit{Ş. Konca}, Topology Appl. 275, Article ID 106996, 9 p. (2020; Zbl 1437.54020) Full Text: DOI References: [1] Arhangel’skii, A. V., An extremal disconnected bicompactum of weight c is inhomogeneous, Dokl. Akad. Nauk SSSR, 175, 751-754 (1967) [2] Arhangel’skii, A. 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