## Hereditary topological diagonalizations and the Menger-Hurewicz Conjectures.(English)Zbl 1137.54018

This paper deals with selection/diagonalization properties stating that from the terms of a sequence of covers of type $$\mathcal{A}$$ one may select small (one-element/finite) subfamilies to form a cover of type $$\mathcal{B}$$. The authors deal with four types of (countable) covers: covers, $$\omega$$-covers (each finite subset is in some element of the cover), $$\tau$$-covers (each element is in infinitely many elements of the cover and if $$x\neq y$$ then one of $$x\in U\not\ni y$$ or $$x\notin U\ni y$$ holds for only finitely many $$U$$) and $$\gamma$$-covers (each point is in all but finitely many members of the cover).
In this paper the authors investigate whether for open and Borel covers these properties are hereditary. A fair number of Borel properties are hereditary; this is because a countable cover by Borel sets of a subspace is easily augmented (using the complement of its union) to form a countable Borel cover of the ambient space.
From $$\mathfrak{p}=\mathfrak{c}$$ the authors construct a set of reals with the property that for every sequence of $$\omega$$-covers there is a choice function that forms a $$\gamma$$-cover, yet when the rationals are removed one obtains a space with a sequence $$\langle \mathcal{U}_n\rangle_n$$ of $$\gamma$$-covers such that for no finite choices $$\mathcal{F}_n \subseteq \mathcal{U}_n$$ the family $$\{\bigcup\mathcal{F}_n\}_n$$ is a cover. This shows that none of the properties studied in the paper is provably hereditary.
Reviewer: K. P. Hart (Delft)

### MSC:

 54D20 Noncompact covering properties (paracompact, Lindelöf, etc.) 54G20 Counterexamples in general topology 54A35 Consistency and independence results in general topology
Full Text:

### References:

 [1] Tomek Bartoszynski, Remarks on small sets of reals, Proc. Amer. Math. Soc. 131 (2003), no. 2, 625 – 630. · Zbl 1017.03027 [2] Tomek Bartoszynski and Saharon Shelah, Continuous images of sets of reals, Topology Appl. 116 (2001), no. 2, 243 – 253. · Zbl 0992.03061 [3] Tomek Bartoszynski, Saharon Shelah, and Boaz Tsaban, Additivity properties of topological diagonalizations, J. Symbolic Logic 68 (2003), no. 4, 1254 – 1260. · Zbl 1071.03031 [4] A. R. Blass, Combinatorial cardinal characteristics of the continuum, in: Handbook of Set Theory , Kluwer Academic Publishers, Dordrecht, to appear. · Zbl 1198.03058 [5] J. Chaber and R. Pol, A remark on Fremlin-Miller theorem concerning the Menger property and Michael concentrated sets, preprint. [6] Eric K. van Douwen, The integers and topology, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 111 – 167. · Zbl 0561.54004 [7] Arnold W. Miller and David H. Fremlin, On some properties of Hurewicz, Menger, and Rothberger, Fund. Math. 129 (1988), no. 1, 17 – 33. · Zbl 0665.54026 [8] Fred Galvin and Arnold W. Miller, \?-sets and other singular sets of real numbers, Topology Appl. 17 (1984), no. 2, 145 – 155. · Zbl 0551.54001 [9] J. Gerlits and Zs. Nagy, Some properties of \?(\?). I, Topology Appl. 14 (1982), no. 2, 151 – 161. · Zbl 0503.54020 [10] W. Hurewicz, Über eine Verallgemeinerung des Borelschen Theorems, Mathematische Zeitschrift 24 (1925), 401-421. · JFM 51.0454.02 [11] Winfried Just, Arnold W. Miller, Marion Scheepers, and Paul J. Szeptycki, The combinatorics of open covers. II, Topology Appl. 73 (1996), no. 3, 241 – 266. , https://doi.org/10.1016/S0166-8641(96)00075-2 Marion Scheepers, Combinatorics of open covers. III. Games, \?_{\?}(\?), Fund. Math. 152 (1997), no. 3, 231 – 254. · Zbl 0870.03021 [12] Ljubiša D. R. Kočinac and Marion Scheepers, Combinatorics of open covers. VII. Groupability, Fund. Math. 179 (2003), no. 2, 131 – 155. · Zbl 1115.91013 [13] M. K. Menger, Einige Überdeckungssätze der Punktmengenlehre, Sitzungsberichte der Wiener Akademie 133 (1924), 421-444. · JFM 50.0129.01 [14] Arnold W. Miller, On the length of Borel hierarchies, Ann. Math. Logic 16 (1979), no. 3, 233 – 267. · Zbl 0415.03038 [15] Arnold W. Miller, A nonhereditary Borel-cover \?-set, Real Anal. Exchange 29 (2003/04), no. 2, 601 – 606. · Zbl 1069.03047 [16] Andrej Nowik, Marion Scheepers, and Tomasz Weiss, The algebraic sum of sets of real numbers with strong measure zero sets, J. Symbolic Logic 63 (1998), no. 1, 301 – 324. · Zbl 0901.03036 [17] Ireneusz Recław, Every Lusin set is undetermined in the point-open game, Fund. Math. 144 (1994), no. 1, 43 – 54. · Zbl 0809.04002 [18] Marion Scheepers, Combinatorics of open covers. I. Ramsey theory, Topology Appl. 69 (1996), no. 1, 31 – 62. · Zbl 0848.54018 [19] Marion Scheepers, Sequential convergence in \?_{\?}(\?) and a covering property, East-West J. Math. 1 (1999), no. 2, 207 – 214. · Zbl 0976.54016 [20] Marion Scheepers and Boaz Tsaban, The combinatorics of Borel covers, Topology Appl. 121 (2002), no. 3, 357 – 382. · Zbl 1025.03042 [21] J. Steprans, Question about which I have thought, preprint: http://www.math.yorku.ca/Who/Faculty/Steprans/Research/q.pdf [22] Boaz Tsaban, A topological interpretation of \?, Real Anal. Exchange 24 (1998/99), no. 1, 391 – 404. · Zbl 0938.03071 [23] Boaz Tsaban, A diagonalization property between Hurewicz and Menger, Real Anal. Exchange 27 (2001/02), no. 2, 757 – 763. · Zbl 1044.26001 [24] B. Tsaban, Selection principles and the minimal tower problem, Note di Matematica 22 (2003), 53-81. · Zbl 1176.03022 [25] B. Tsaban, Strong $$\gamma$$-sets and other singular spaces, Topology and its Applications, to appear. · Zbl 1094.54010 [26] B. Tsaban and T. Weiss, Products of special sets of real numbers, Real Analysis Exchange, to appear. · Zbl 1103.03043 [27] Piotr Zakrzewski, Universally meager sets, Proc. Amer. Math. Soc. 129 (2001), no. 6, 1793 – 1798. · Zbl 0967.03043
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.