Filomat 33:7 (2019), 2061–2071 https://doi.org/10.2298/FIL1907061K Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Compact Complement Topologies and k-Spaces K. Keremedis a , C. Özel b , A. Pi ˛ ekosz c , M.A. Al Shumrani b , E. Wajch d a Department of Mathematics, University of Aegean, Karlovassi, Samos 83200, Greece b Department of Mathematics, King Abdulaziz University, P.O.Box: 80203 Jeddah 21589, Saudi Arabia c Institute of Mathematics, Cracow University of Technology, Warszawska 24, 31-155 Kraków, Poland d Siedlce University of Natural Sciences and Humanities, 3 Maja 54, 08-110 Siedlce, Poland Abstract. Let (X,τ) be a Hausdorff space, where X is an infinite set. The compact complement topology τ ⋆ on X is defined by: τ ⋆ = {∅} ∪ {X \ M : M is compact in (X,τ)}. In this paper, properties of the space (X,τ ⋆ ) are studied in ZF and applied to a characterization of k-spaces, to the Sorgenfrey line, to some statements independent of ZF, as well as to partial topologies that are among Delfs-Knebusch generalized topologies. Between other results, it is proved that the axiom of countable multiple choice (CMC) is equivalent with each of the following two sentences: (i) every Hausdorff first-countable space is a k-space, (ii) every metrizable space is a k-space. A ZF-example of a countable metrizable space whose compact complement topology is not first-countable is given. 1. Introduction The compact complement topology of the real line was considered, for instance, in Example 22 of the celebrated book by Steen and Seebach "Counterexamples in Topology" ([19]). We investigate this notion in a much wider context of Hausdorff spaces and of partially topological spaces that belong to the class of generalized topological spaces in the sense of Delfs-Knebusch (cf. [2] and [14]). Our results are proved in ZF if this is not otherwise stated. All axioms of ZF can be found in [11]. In Section 2, we give elementary properties of the compact complement topology of a Hausdorff space. In particular, we show that if a Hausdorff space is locally compact and second-countable, then its compact complement topology is second-countable, while the compact complement topology of a non- locally compact metrizable space need not be first-countable. We give an example of a countable metrizable space whose compact complement topology is not first-countable. In Section 3, a necessary and sufficient condition for a set to be compact with respect to the compact complement topology of a given Hausdorff space leads us to a new characterization of k-spaces. A well-known theorem of ZFC states that all first- countable Hausdorff spaces are k-spaces (cf. Theorem 3.3.20 of [3]). We show that this theorem may fail in ZF. More precisely, we prove that, if M is a model of ZF, then all Hausdorff first-countable spaces of M are k-spaces if and only if all metrizable spaces in M are k-spaces which holds if and only if the axiom of 2010 Mathematics Subject Classification. Primary: 54D50; Secondary: 54A35, 54D30, 54D55, 54E25, 54E35, 54E99 Keywords. Compact complement topology, countable multiple choice, k-space, sequential space, Sorgenfrey line, Delfs-Knebusch generalized topological space, partial topology Received: 02 July 2018; Revised: 29 November 2018; Accepted: 05 December 2018 Communicated by Ljubiša D.R. Koˇ cinac Email addresses: kker@aegean.gr (K. Keremedis), cenap.ozel@gmail.com (C. Özel), pupiekos@cyfronet.pl (A. Pi ˛ ekosz), maalshmrani1@kau.edu.sa (M.A. Al Shumrani), eliza.wajch@wp.pl (E. Wajch)