Products of compact spaces and the axiom of choice II (original) (raw)
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
Background: For infinite products of compact spaces, Tychonoff's theorem asserts that their product is compact, in the product topology. Tychonoff's theorem is shown to be equivalent to the axiom of choice. In this paper, we show that any countable product of compact metric spaces iscompact, without using Tychonoff's theorem. The proof needs only basic and standard facts of compact metric spaces and the Bolzano-Weierstrass property. Moreover, the component spaces need not be assumed to be copies of the same compact metric space, and each component space can be an arbitrary nonempty compact metric space independently. Materials and Methods: Total boundedness together with completeness of a metric space implies its compactness. Completeness of a product of complete spaces is easily inferred from the completeness of each component. Total boundedness therefore suffices to prove the compactness of a countably (infinitely) many nonempty compact component spaces. The countable infiniteness is needed in the proof to exhibit a standard metric that gives rise to the product topology.Any such metric topology for the product arises as exhibited, andthey are all equivalent to the product topology. The requirement of summability ofthe sequences restricts the scope of the result to countably infinite products. Results: The product space obtained by taking the product of any sequence of nonempty compact metric spaces in the product topology is shown to be compact, using only the basic and standard facts of compact metric spaces.Conclusion: Compactness of the product of a countably infinitely many nonempty compact metric spaces can be proved within Cantor's set theory, without using the axiom of choice and Tychonoff's theorem.
Spaces with compact topologies
Rocky Mountain Journal of Mathematics, 1972
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.
A Study on Compactness in Metric Spaces and Topological Spaces
Pure and Applied Mathematics Journal, 2014
Topology may be considered as an abstract study of the limit point concept. As such, it stems in part from recognition of the fact that many important mathematical topics depend entirely upon the properties of limit points. This study shows that compactness implies limit point compactness but not conversely and every compact space is locally compact but not conversely. This study also shows that compactness, limit point compactness and sequentially compactness are equivalent in metrizable spaces and the product of finitely many compact spaces is a locally compact space. This study introduce it here as an interesting application of the Tychonoff theorem.
$K$-Spaces, Sequential Spaces and Related Topics in the Absence of the Axiom of Choice
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.
More on countably compact, locally countable spaces
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.
On Countably alpha\alphaalpha-Compact Topological Spaces
arXiv (Cornell University), 2022
In this paper, some features of countably α-compact topological spaces are presented and proven. The connection between countably α-compact, Tychonoff, and α-Hausdorff spaces is explained. The space is countably α-compact space iff every locally finite family of non-empty subsets of such space is finite is demonstrated. The countably α-compact space with weight greater than or equal to ℵ 0 is the α-continuous image of a closed subspace of the cube D ℵ0 is discussed. The boundedness of α-continuous functions mapping α-compact spaces to other spaces is cleared. Moreover, the α-continuous function mapping the space X to the countably α-compact space Y is an α-closed subset of X × Y is argued and proved. We explained that the α-continuous functions mapping any topological space to a countably α-compact space can be extended over its domain under some constraints. We claimed that the property of being α-compact is countably α-compact but the converse is not and the countable union of countably α-compact subspaces of X is also countably α-compact.
Second-countable compact Hausdorff spaces as remainders in ZF and two new notions of infiniteness
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.
Tychonoff-like Product Theorems for Local Topological Properties
2013
We consider classes T of topological spaces (referred to as T-spaces) that are stable under continuous images and frequently under arbitrary products. A local T-space has for each point a neighborhood base consisting of subsets that are T-spaces in the induced topology. A general necessary and sufficient criterion for a product of topological spaces to be a local T-space in terms of conditions on the factors enables one to establish a broad variety of theorems saying that a product of spaces has a certain local property (like local compactness, local sequential compactness, local σ-compactness, local connectedness etc.) if and only if each factor has that local property, almost all have the corresponding global property, and not too many factors fail a suitable additional condition. Many of the results admit a point-free formulation; a look at sum decompositions into components of spaces with local properties yields product decompositions into indecomposable factors for certain classes of frames like completely distributive lattices or hypercontinuous frames.
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 mathbfZF\mathbf{ZF}mathbfZF. 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 mathbfZFC\mathbf{ZFC}mathbfZFC, is proved to be independent of mathbfZF\mathbf{ZF}mathbfZF. Urysohn's Metrization Theorem is generalized to the following theorem: every T_3T_3T_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 mathbfZF\mathbf{ZF}mathbfZF 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...
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
On first countable, cellular-compact spaces
arXiv (Cornell University), 2019
As it was introduced by Tkachuk and Wilson in , a topological space X is cellular-compact if for any cellular, i.e. disjoint, family U of nonempty open subsets of X there is a compact subspace K ⊂ X such that In this note we answer several questions raised in by showing that (1) any first countable cellular-compact T 2 space is T 3 , and so its cardinality is at most c = 2 ω ; (2) cov(M) > ω 1 implies that every first countable and separable cellularcompact T 2 space is compact; (3) if there is no S-space then any cellular-compact T 3 space of countable spread is compact; (4) M Aω 1 implies that every point of a compact T 2 space of countable spread has a disjoint local π-base.
The axiom of choice in topology
Notre Dame Journal of Formal Logic, 1983
In this paper we are concerned with soft applications of the axiom of choice {AC) in general topology. We define 16 properties which hold in ZF for each T 2 space, if and only if AC is true, and we investigate what implications between these axioms are provable without AC (in the presence of AC there is nothing to prove). Our results are summarized in two diagrams. In Figure 1 the 61 valid implications are listed. Counterexamples in three models prove 188 of the possible implications to depend on the axiom of choice, as is shown in Figure 2. Some problems remain open. Our positive results are proved in ZF°, Zermelo-Fraenkel set theory without the axioms AC and foundation. Our counterexamples are constructed in models of ZF (=^ ZF° + foundation). We shall use Levy's axiom MC of multiple choice: If F is a family of nonempty sets, there is a mapping /on F, such that φ Φ fix) C x and f{x) is finite for each x in F. Rubin's axiom PW asserts that the power set i°{x) of each well-orderable set x is well-orderable. In ZF°, AC implies MC which implies PW, and there are permutation models which show that in ZF° the implications cannot be reversed. But AC, MC, and PW are equivalent in ZF. Similar axioms are studied in [6]. If P and Q are topological properties, A(P) is the assertion that each T 2 space is P and A{P, Q) says that each T 2 space which is P also satisfies Q. While for the properties P defined below we do understand the position of A(P) in the hierarchy of choice principles, the same questions for A{P,Q) remain unanswered in many cases, although as we have already noticed A{P,Q) depends on AC in general. *The author wishes to express his gratitude to the referee and to U. Feigner (Tubingen) for their many useful suggestions.
κ-compactness, extent and the Lindelöf number in LOTS
Central European Journal of Mathematics, 2014
We study the behaviour of ℵ-compactness, extent and Lindelöf number in lexicographic products of linearly ordered spaces. It is seen, in particular, that for the case that all spaces are bounded all these properties behave very well when taking lexicographic products. We also give characterizations of these notions for generalized ordered spaces.
Fundamentals of Contemporary Mathematical Sciences
In a recent paper, a novel class of generalized compact sets (briefly, g-Tg -compact sets) in generalized topological spaces (briefly, Tg -spaces) has been studied. In this paper, the concept is further studied and, other derived concepts called countable, sequential, and local generalized compactness (countable, sequential, local g-Tg -compactness) in Tg -spaces are also studied relatively. The study reveals that g-Tg -compactness implies local g-Tg -compactness and countable g-Tg-compactness, sequential g-Tg -compactness implies countable g-Tg -compactness and, g-Tg -compactness is a generalized topological property (briefly, Tg -property). Diagrams establish the various relationships amongst these types of g-Tg -compactness presented here and in the literature, and a nice application supports the overall theory.
TWO TOPOLOGICAL EQUIVALENTS OF THE AXIOM OF CHOICE
Mathematical Logic Quarterly, 1992
We show that the Axiom of Choice is equivalent to each of the following statements: (i) A product of closures of subsets of topological spaces is equal to the closure of their product (in the product topology); (ii) A product of complete uniform spaces is complete.
Compact covers and function spaces
Journal of Mathematical Analysis and Applications, 2014
For a Tychonoff space X, we denote by C p (X) and C c (X) the space of continuous real-valued functions on X equipped with the topology of pointwise convergence and the compact-open topology respectively. Providing a characterization of the Lindelöf Σ-property of X in terms of C p (X), we extend Okunev's results by showing that if there exists a surjection from C p (X) onto C p (Y) (resp. from L p (X) onto L p (Y)) that takes bounded sequences to bounded sequences, then υY is a Lindelöf Σ-space (respectively K-analytic) if υX has this property. In the second part, applying Christensen's theorem, we extend Pelant's result by proving that if X is a separable completely metrizable space and Y is first countable, and there is a quotient linear map from C c (X) onto C c (Y), then Y is a separable completely metrizable space. We study also the non-separable case, and consider a different approach to the result of J. Baars, J. de Groot, J. Pelant and V. Valov, which is based on the combination of two facts: Complete metrizability is preserved by p-equivalence in the class of metric spaces (J. Baars, J. de Groot, J. Pelant). If X is completely metrizable and p-equivalent to a first countable Y , then Y is metrizable (V. Valov). Some additional results are presented.