Bella, Angelo; Spadaro, Santi A common extension of Arhangel’skĭ’s theorem and the Hajnal-Juhász inequality. (English) Zbl 1437.54004 Can. Math. Bull. 63, No. 1, 197-203 (2020). For a space \(X\) the piecewise weak Lindelöf degree for closed sets of \(X\), pwL\(_c(X)\), is the minimum cardinal \(\kappa\) such that for any closed \(C\subset X\), any open cover \(\mathcal U\) of \(C\) and any decomposition \(\langle\mathcal U_i\rangle_{i\in I}\) of \(\mathcal U\) there are subfamilies \(\mathcal V_i\subset\mathcal U_i\) of size at most \(\kappa\) such that \(C\subset\cup_{i\in I}\left(\overline{\cup\mathcal V_i}\right)\). It is shown that \(|X|\le2^{\mathrm{pwL}_c(X)\chi(X)}\) when \(X\) is Hausdorff. Since pwL\(_c(X)\le\min(L(X),c(X))\), this gives a common extension of inequalities of Arhangel’skǐ and Hajnal-Juhász. It is noted that this inequality is a strict improvement on the other two: if \(X=(S\times S)\oplus A([0,1])\), where \(S\) is the Sorgenfrey line and \(A([0,1])\) the Aleksandroff duplicate of \([0,1]\), then \(\mathrm{pwL}_c(X)\chi(X)=\aleph_0\) while \(L(X)=c(X)=\mathfrak c\). Reviewer: David B. Gauld (Auckland) Cited in 1 ReviewCited in 9 Documents MSC: 54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets) 54D20 Noncompact covering properties (paracompact, Lindelöf, etc.) 54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.) Keywords:cardinality bound; cardinal invariant; cellularity; Lindelöf; weakly Lindelöf; piecewise weakly Lindelöf PDFBibTeX XMLCite \textit{A. Bella} and \textit{S. Spadaro}, Can. Math. Bull. 63, No. 1, 197--203 (2020; Zbl 1437.54004) Full Text: DOI arXiv References: [1] Alas, O. T., More topological cardinal inequalities. Colloq. Math.65(1993), 165-168. · Zbl 0842.54002 [2] Arhangel’skiĭ, A. V., The power of bicompacta with first axiom of countability. Soviet Math. Dokl.10(1969), 951-955. [3] Arhangel’skiĭ, A. V., A theorem about cardinality. Russian Math. Surveys34(1979), 153-154. [4] Arhangel’skiĭ, A. V., A generic theorem in the theory of cardinal invariants of topological spaces. Comment. Math. Univ. Carolin.36(1995), 303-325. [5] Bell, M., Ginsburg, J. N., and Woods, R. G., Cardinal inequalities for topological spaces involving the weak Lindelöf number. Pacific J. Math.79(1978), 37-45. · Zbl 0367.54003 [6] Bella, A. and Cammaroto, F., On the cardinality of Urysohn spaces. Canad. Math. Bull.31(1988), 153-158. · Zbl 0646.54005 [7] Bella, A. and Carlson, N., On cardinality bounds involving the weak Lindelöf degree. Quaest. Math.41(2018), 99-113. · Zbl 1429.54028 [8] Bella, A. and Spadaro, S., Cardinal invariants for the G_𝛿-topology. Colloq. Math.156(2019), 123-133. · Zbl 1420.54007 [9] Bella, A. and Spadaro, S., Infinite games and cardinal properties of topological spaces. Houston J. Math.41(2015), 1063-1077. · Zbl 1343.54003 [10] Dow, A., An introduction to applications of elementary submodels to topology. Topology Proc.13(1988), 17-72. · Zbl 0696.03024 [11] Engelking, R., General topology. Heldermann-Verlag, 1989. · Zbl 0684.54001 [12] Gotchev, I., Cardinalities of weakly Lindelöf spaces with regular G_𝜅 diagonals. Topology Appl.259(2019), 80-89. · Zbl 1426.54009 [13] Hajnal, A. and Juhász, I., Discrete subspaces of topological spaces. Indag. Math.29(1967), 343-356. · Zbl 0163.17204 [14] Hodel, R. E., Cardinal Functions I. In: Handbook of set-theoretic topology, eds. K. Kunen and J. E. Vaughan, North Holland, Amsterdam, 1984, pp. 1-61. [15] Hodel, R. E., Arhangel’skĭ’s solution to Alexandroff’s problem: A survey. Topology Appl.153(2006), 2199-2217. · Zbl 1099.54001 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.