Tkachuk, Vladimir V. Reflecting topological properties in continuous images. (English) Zbl 1257.54022 Cent. Eur. J. Math. 10, No. 2, 456-465 (2012). Let \(\kappa\) be an infinite cardinal. A cardinal invariant \(\eta\) is said to reflect in continuous images of weight \(\leq \kappa^+\) if for any space \(X\) to satisfy \(\eta(X)\leq\kappa\) it is enough that \(\eta(Y)\leq\kappa\) for every continuous image \(Y\) of \(X\) with weight \(\leq \kappa^+\). Analogously, a topological property \(\mathcal P\) is said to reflect in continuous images of weight \(\leq \kappa^+\) if the following are equivalent for any space \(X\): (i) \(X\) has \(\mathcal P\). (ii) Every continuous image of \(X\) with weight less than or equal to \(\kappa^+\) has \(\mathcal P\).From the Introduction: “We establish that, for any infinite cardinal \(\kappa\), the spread, character, pseudocharacter and Souslin number reflect in continuous images of weight \(\kappa^+\) for arbitrary Tychonoff spaces; the same is true for network weight under the hypothesis \(2^{\kappa}=\kappa^+\). We also show that tightness reflects in continuous images of weight \(\kappa^+\) for compact spaces. We present examples showing that separability, countable extent and normality do not reflect in continuous images of weight \(\omega_1.\)”Further results are proved under additional axioms of set theory and a number of examples and open questions are provided. Reviewer: Xabier Domínguez (La Coruña) Cited in 5 Documents MSC: 54C05 Continuous maps 54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets) Keywords:continuous images; reflected properties; small weight; character; pseudocharacter; Souslin number; network weight PDFBibTeX XMLCite \textit{V. V. Tkachuk}, Cent. Eur. J. Math. 10, No. 2, 456--465 (2012; Zbl 1257.54022) Full Text: DOI References: [1] Alas O.T., Tkachuk V.V., Wilson R.G., Closures of discrete sets often reflect global properties, Topology Proc., 2000, 25(Spring), 27-44; · Zbl 1002.54021 [2] Arkhangel’skił A.V., Continuous mappings, factorization theorems and spaces of functions, Trudy Moskov. Mat. Obshch., 1984, 47, 3-21 (in Russian); [3] Arkhangel’skił A.V., Topological Function Spaces, Kluwer, Dordrecht, 1992 http://dx.doi.org/10.1007/978-94-011-2598-7; [4] Dow A., An empty class of nonmetric spaces, Proc. Amer. Math. Soc., 1988, 104(3), 999-1001 http://dx.doi.org/10.1090/S0002-9939-1988-0964886-9; · Zbl 0692.54018 [5] Engelking R., General Topology, Mathematical Monographs, 60, PWN, Warsaw, 1977; [6] Gul’ko S.P., Properties of sets that lie in Σ-products, Dokl. Akad. Nauk SSSR, 1977, 237(3), 505-508 (in Russian); [7] Hajnal A., Juhász I., Having a small weight is determined by the small subspaces, Proc. Amer. Math. Soc., 1980, 79(4), 657-658 http://dx.doi.org/10.1090/S0002-9939-1980-0572322-2; · Zbl 0432.54003 [8] Juhász I., Consistency results in topology, In: Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977, 503-522 http://dx.doi.org/10.1016/S0049-237X(08)71112-1; [9] Juhász I., Cardinal Functions in Topology — Ten Years Later, Math. Centre Tracts, 123, Mathematisch Centrum, Amsterdam, 1980; · Zbl 0479.54001 [10] Kalenda O.F.K., Note on countable unions of Corson countably compact spaces, Comment. Math. Univ. Carolin., 2004, 45(3), 499-507; · Zbl 1098.54020 [11] Ramírez-Páramo A., A reflection theorem for i-weight, Topology Proc., 2004, 28(1), 277-281; · Zbl 1079.54005 [12] Tkachenko M.G., Continuous mappings onto spaces of smaller weight, Moscow Univ. Math. Bull., 1980, 35(2), 41-44; · Zbl 0459.54003 [13] Tkachuk V.V., Spaces that are projective with respect to classes of mappings, Trans. Moscow Math. Soc., 1988, 139-156; · Zbl 0662.54007 [14] Tkachuk V.V., A short proof of a classical result of M.G. Tkachenko, Topology Proc., 2001/02, 26(2), 851-856; [15] Tkachuk V.V., A C p-Theory Problem Book, Springer, New York-Dordrecht-Heidelberg-London, 2011 http://dx.doi.org/10.1007/978-1-4419-7442-6; 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.