CH, a problem of Rolewicz and bidiscrete systems

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.

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

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

Transactions of the American Mathematical Society, 2020

The paper elucidates the relationship between the density of a Banach space and possible sizes of Auerbach systems and well-separated subsets of its unit sphere. For example, it is proved that for a large enough space X X , the unit sphere S X S_X always contains an uncountable ( 1 + ) (1+) -separated subset. In order to achieve this, new results concerning the existence of large Auerbach systems are established, that happen to be sharp for the class of weakly Lindelöf determined (WLD) spaces. In fact, we offer the first consistent example of a non-separable WLD Banach space that contains no uncountable Auerbach system, as witnessed by a renorming of c 0 ( ω 1 ) c_0(\omega _1) . Moreover, the following optimal results for the classes of, respectively, reflexive and super-reflexive spaces are established: the unit sphere of an infinite-dimensional reflexive space contains a symmetrically ( 1 + ε ) (1+\varepsilon ) -separated subset of any regular cardinality not exceeding the density...

Canadian Journal of Mathematics, 1976

Introduction. Our method using CH is a blend of two earlier constructions and Ostaszewski ) of hereditarily separable {HS), regular, non-Lindelôf, first countable spaces. [4] produces a much better space than ours in § 1 ; it has all of our properties except that it is not realcompact (which is probably more interesting), and it is countably compact as well; however, the construction works only under O, which implies the continuum hypothesis (CH) but is not equivalent to it. The argument of [2], like ours, just needs CH, but it is much more complicated, and it is not immediate that the space produced is locally compact or perfectly normal (although, in fact, it is; see the remark at the end of § 1). In § 2, we use a more complicated version of the technique in § 1 to construct a first countable, cardinality coi, HS, Dowker space. A Dowker space is a normal, Hausdorff space which is not countably paracompact. There is a known "real" Dowker space but all of its cardinal functions are large . There is a known HS Dowker space but its construction depends on the existence of a Souslin line . It was an old conjecture that the existence of a small cardinality (or small cardinal function) Dowker space depended on the existence of a Souslin line, and this conjecture is disproved by our construction. Using our technique and O (which implies both CH and the existence of a Souslin line) we can construct a first countable, cardinality coi, HS, Dowker space which is also locally compact and c-countably compact; but we choose the weaker hypothesis over the stronger conclusion. In § 2 we use Lusin sets in our construction. A subset L of the line is Lusin if L is uncountable and every nowhere dense subset of L is countable. If we assume CH, then there are Lusin sets in the line. However if we assume Martin's axiom and the negation of CH, then there are no Lusin sets in the line. If we assume Martin's axiom and the negation of CH, then there is no non-Lindelôf, first countable, regular topology on a subset of the line which refines the usual topology and has the property that the closure of a set in the two topologies differs by an at most countable set. Since our construction in § 1 yields just such a topology, both constructions are independent of the usual axioms for set theory. 1. The basic idea for obtaining this space is to start with the usual topology of the real numbers (R), which has many of the properties we want; in particular

International Journal of Mathematics and Mathematical Sciences, 2010

In 1972, Bennett studied the countable dense homogeneous (CDH) spaces and in 1992, Fitzpatrick, White, and Zhou proved that every CDH space is aT1space. Afterward Bsoul, Fora, and Tallafha gave another proof for the same result, also they defined the almost CDH spaces and almostT1,T0spaces, indeed they prove that every ACDH space is an almostT1space. In this paper we introduce a new type of almost CDH spaces called ACDH-1, we characterize the ACDH spaces, the almostT0spaces, we also give relations between different types of CDH spaces. We define new type of almostT1(AT1) spaces, and we study the relations between the old and new definitions. By extending the techniques given by Tallafha, Bsoul, and Fora, we prove that every ACDH-1 is anAT1.

Let X be a Banach space with an uncountable unconditional Schauder basis, and let Y be an arbitrary nonseparable subspace of X. If X contains no isomorphic copy of 1(J ) with J uncountable then (1) the density of Y and the weak*-density of Y * are equal, and (2) the unit ball of X * is weak* sequentially compact. Moreover, (1) implies that Y contains large subsets consisting of pairwise disjoint elements, and a similar property holds for uncountable unconditional basic sets in X.

Czechoslovak Academy of Sciences(Praha), 1980

2019

We first introduce and study two new classes of subsets in T0 spaces ω-Rudin sets and ω-well-filtered determined sets lying between the class of all closures of countable directed subsets and that of irreducible closed subsets, and two new types of spaces ω-d spaces and ω-well-filtered spaces. We prove that an ω-well-filtered T0 space is locally compact iff it is core compact. One immediate corollary is that every core compact well-filtered space is sober, answering Jia-Jung problem with a new method. We also prove that all irreducible closed subsets in a first countable ω-well-filtered T0 space are directed. Therefore, a first countable T0 space X is sober iff X is well-filtered iff X is an ω-well-filtered d-space. Using ω-well-filtered determined sets, we present a direct construction of the ω-well-filtered reflections of T0 spaces, and show that products of ω-well-filtered spaces are ω-well-filtered.

Fundamenta Mathematicae

We construct two examples of a compact, 0-dimensional space which supports a Radon probability measure whose measure algebra is isomorphic to the measure algebra of 2 ω1. The first construction uses ♦ to produce an S-space with no convergent sequences in which every perfect set is a G δ. A space with these properties must be both hereditarily normal and hereditarily countably paracompact. The second space is constructed under CH and is both HS and HL. 0. Introduction. All spaces considered here are Hausdorff. A perfect set is a non-empty closed set with no isolated points. Suppose X is compact and supports a Radon probability measure µ such that the measure algebra of (X, µ) is not separable; does this imply that X can be mapped continuously onto [0, 1] ω 1 ? This question is open in ZFC. In particular, Haydon asked whether such an implication might follow from something like MA + ¬CH; see Fremlin [2] for more discussion. Under CH, there is a counterexample which is, in addition, a compact Lspace (hereditarily Lindelöf (HL) but not hereditarily separable (HS)); see [4, 6]. In this paper, we show that, assuming ♦, there is another counterexample which is an S-space (HS, but not HL). The space also has the property that every perfect set is a G δ , whereas no point is a G δ. Also, assuming just CH, we construct a third counterexample which is both HS and HL. Neither of the above mentioned examples could be constructed in ZFC, since under MA + ¬CH, there are neither compact L-spaces (Juhász) nor compact S-spaces (Szentmiklóssy) (see [8]). Furthermore, under MA + ¬CH, the measure algebra of any compact HL (equivalently, HS) Radon measure space is separable (Fremlin [2]). The following theorem details the properties of the S-space. The HS + HL example is a modification of the S-space, and is described in §4.