Spaces with compact topologies

Filomat

Let (X,?) be a Hausdorff space, where X is an infinite set. The compact complement topology ?* on X is defined by: ?* = {0}?{X\M:M is compact in (X,?)}. In this paper, properties of the space (X,?*) 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.

2021

In the absence of the axiom of choice, new results concerning sequential, Fréchet-Urysohn, k-spaces, very k-spaces, Loeb and Cantor completely metrizable spaces are shown. New choice principles are introduced. Among many other theorems, it is proved in ZF that every Loeb, T3-space having a base expressible as a countable union of finite sets is a metrizable second-countable space whose every Fσ-subspace is separable; moreover, every Gδ-subspace of a second-countable, Cantor completely metrizable space is Cantor completely metrizable, Loeb and separable. It is also noticed that Arkhangel’skii’s statement that every very k-space is Fréchet-Urysohn is unprovable in ZF but it holds in ZF that every first-countable, regular very k-space whose family of all nonempty compact sets has a choice function is Fréchet-Urysohn. That every second-countable metrizable space is a very k-space is equivalent to the axiom of countable choice for R.

2021

In the absence of the Axiom of Choice, necessary and sufficient conditions for a locally compact Hausdorff space to have all non-empty second-countable compact Hausdorff spaces as remainders are given in ZF. Among other independence results, the characterization of locally compact Hausdorff spaces having all non-empty metrizable compact spaces as remainders, obtained by Hatzenbuhler and Mattson in ZFC, is proved to be independent of ZF. Urysohn's Metrization Theorem is generalized. New concepts of a strongly filterbase infinite set and a dyadically filterbase infinite set are introduced, both stemming from the investigations on compactifications. Set-theoretic and topological definitions of the new concepts are given, and their relationship with certain known notions of infinite sets is investigated in ZF. A new permutation model is introduced in which there exists a strongly filterbase infinite set which is weakly Dedekind-finite.

arXiv: General Topology, 2020

In the absence of the Axiom of Choice, necessary and sufficient conditions for a locally compact Hausdorff space to have all non-empty second-countable compact Hausdorff spaces as remainders are given in $\mathbf{ZF}$. Among other independence results, the characterization of locally compact Hausdorff spaces having all non-empty metrizable compact spaces as remainders, obtained by Hatzenhuhler and Mattson in $\mathbf{ZFC}$, is proved to be independent of $\mathbf{ZF}$. Urysohn's Metrization Theorem is generalized to the following theorem: every $T_3$-space which admits a base expressible as a countable union of finite sets is metrizable. Applications to solutions of problems concerning the existence of some special metrizable compactifications in $\mathbf{ZF}$ are shown. New concepts of a strongly filterbase infinite set and a dyadically filterbase infinite set are introduced, both stemming from the investigations on compactifications. Set-theoretic and topological definitions o...

Mathematical Logic Quarterly, 2003

This is a continuation of [dhhkr]. We study the Tychonoff Compactness Theorem for various definitions of compact and for various types of spaces, (first and second countable spaces, Hausdorff spaces, and subspaces of R κ ). We also study well ordered Tychonoff products and the effect that the multiple choice axiom has on such products.

2021

The main aim of the article is to show, in the absence of the Axiom of Choice, relationships between the following, independent of ZF, statements: “Every countable product of compact metrizable spaces is separable (respectively, compact)” and “Every countable product of compact metrizable spaces is metrizable”. Statements related to the above-mentioned ones are also studied. Permutation models (among them new ones) are shown in which a countable sum (also a countable product) of metrizable spaces need not be metrizable, countable unions

TJPRC, 2014

The aim of this paper is to introduce and study the concepts of -compact space, -compact subspace and countably -compact space via -open sets like wise to investigate their relationships to other well known types of compactness

Monatshefte für Mathematik, 2021

In the absence of the axiom of choice, the set-theoretic status of many natural statements about metrizable compact spaces is investigated. Some of the statements are provable in ZF, some are shown to be independent of ZF. For independence results, distinct models of ZF and permutation models of ZFA with transfer theorems of Pincus are applied. New symmetric models of ZF are constructed in each of which the power set of R is well-orderable, the Continuum Hypothesis is satisfied but a denumerable family of non-empty finite sets can fail to have a choice function, and a compact metrizable space need not be embeddable into the Tychonoff cube [0, 1] R .

Israel Journal of Mathematics, 1988

Following [5], a T 3 space X is called good (splendid) if it is countably compact, locally countable (and ω-fair). G(κ) (resp. S(κ)) denotes the statement that a good (resp. splendid) space X with |X| = κ exists. We prove here that (i) Con(ZF) → Con(ZFC + MA + 2 ω is big + S(κ) holds unless ω = cf (κ) < κ; (ii) a supercompact cardinal implies Con(ZFC + MA + 2 ω > ω ω+1 + ¬G(ω ω+1); (iii) the "Chang conjecture" (ω ω+1 , ωω) → (ω +1, ω) implies ¬S(κ) for all κ ≥ ωω; (iv) if P adds ω 1 dominating reals to V iteratively then, in V P , we have G(λ ω) for all λ.

Topology and its Applications, 2004

We study properties related to first countability and countable compactness of uniformizable ∆hit-and-miss hyperspace topologies. We show that they are proximal hyperspace topologies. We use the Smirnov compactification as a tool in our investigation.