On countable choice and sequential spaces

Under the axiom of choice, every first countable space is a Fréchet‐Urysohn space. Although, in its absence even ? may fail to be a sequential space. Our goal in this paper is to discuss under which set‐theoretic conditions some topological classes, such as the first countable spaces, the metric spaces, or the subspaces of ?, are classes of Fréchet‐Urysohn or sequential spaces. In this context, it is seen that there are metric spaces which are not sequential spaces. This fact raises the question of knowing if the completion of a metric space exists and it is unique. The answer depends on the definition of completion. Among other results it is shown that: every first countable space is a sequential space if and only if the axiom of countable choice holds, the sequential closure is idempotent in ? if and only if the axiom of countable choice holds for families of subsets of ?, and every metric space has a unique ‐completion. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Products of compact spaces and the axiom of choice II

This is a continuation of [2]. We study the Tychonoff Compactness Theorem for various definitions of compactness and for various types of spaces (first and second countable spaces, Hausdorff spaces, and subspaces of ?^K). We also study well ordered Tychonoff products and the effect that the multiple choice axiom has on such products.     

极大点空间的若干特征定理

本文利用极大点空间的等价刻划证明了极大点空间的某些子空间、不交和、 乘积空间、逆序列的逆极限、具有可数基的局部紧的Hausdorff空间是极大点空间,还 给出了具有可数基的局部紧的Hausdorff空间的Domain hull.     

Real numbers and other completions

A notion of completeness and completion suitable for use in the absence of countable choice is developed. This encompasses the construction of the real numbers as well as the completion of an arbitrary metric space. The real numbers are characterized as a complete Archimedean Heyting field, a terminal object in the category of Archimedean Heyting fields. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Long Borel hierarchies

We show that there is a model of ZF in which the Borel hierarchy on the reals has length ω₂. This implies that ω₁ has countable cofinality, so the axiom of choice fails very badly in our model. A similar argument produces models of ZF in which the Borel hierarchy has exactly λ + 1 levels for any given limit ordinal λ less than ω₂. We also show that assuming a large cardinal hypothesis there are models of ZF in which the Borel hierarchy is arbitrarily long. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Unions and the axiom of choice

We study statements about countable and well‐ordered unions and their relation to each other and to countable and well‐ordered forms of the axiom of choice. Using WO as an abbreviation for “well‐orderable”, here are two typical results: The assertion that every WO family of countable sets has a WO union does not imply that every countable family of WO sets has a WO union; the axiom of choice for WO families of WO sets does not imply that the countable union of countable sets is WO. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On quotients of non‐Archimedean Fréchet spaces

It is proved when a non‐Archimedean Fréchet space E of countable type has a quotient isomorphic to ??^?, c^?₀ or c₀ × ??^?. It is also shown when E has a non‐normable quotient with a continuous norm. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

关于可数点集的盒维数的一些探讨

本文研究了一类可数点集的盒维数的计算问题.通过构造双Lipschitz映射,把原可数点集的盒维数的求解问题转化为求解一类相对简单的可数点集的盒维数.获得了两个单调的可数点集在具有同阶间隔时具有相同的上盒维数和下盒维数的结论.该结论为计算一类可数点集的盒维数提供了方便.     

Weak axioms of choice for metric spaces

In the framework of ZF, i.e., Zermelo-Fraenkel set theory without the axiom of choice AC, we show that if the family of all non-empty, closed subsets of a metric space has a choice function, then so does the family of all non-empty, open subsets of . In addition, we establish that the converse is not provable in ZF.

We also show that the statement ``every subspace of the real line with the standard topology has a choice function for its family of all closed, non-empty subsets" is equivalent to the weak choice form ``every continuum sized family of non-empty subsets of reals has a choice function".

     

Countably Complementable Linear Orderings

We say that a countable linear ordering is countably complementable if there exists a linear ordering , possibly uncountable, such that for any countable linear ordering , does not embed into if and only if embeds into . We characterize the linear orderings which are countably complementable. We also show that this property is equivalent to the countable version of the finitely faithful extension property introduced by Hagendorf. Using similar methods and introducing the notion of weakly countably complementable linear orderings, we answer a question posed by Rosenstein and prove the countable case of a conjecture of Hagendorf, namely, that every countable linear ordering satisfies the countable version of the totally faithful extension property. This research was partially supported by NSF grant DMS-0600824.     

Products with linear and countable type factors

The basic theorem presented shows that the product of a linearly ordered space and a countable (regular) space is normal. We prove that the countable space can be replaced by any of a rather large class of countably tight spaces. Examples are given to prove that monotone normality cannot replace linearly ordered in the base theorem. However, it is shown that the product of a monotonically normal space and a monotonically normal countable space is normal.

     

群元值测度的一个定义和一些性质

文中给出了定义在第二可数完备布尔代数上,取值于半序交换群内的群元值测度的一个定义和一些性质.     

Local rings of countable Cohen-Macaulay type

We prove (the excellent case of) Schreyer's conjecture that a local ring with countable CM type has at most a one-dimensional singular locus. Furthermore, we prove that the localization of a Cohen-Macaulay local ring of countable CM type is again of countable CM type.

     

关于具有局部可数sn-网的空间

In this paper,spaces with a locally countable sn-network are discussed.It is shown that a space with a locally countable sn-network iff it is an snf-countable space with a locally countable k-network.As its application,almost-open and closed mappings(or finite-to-one and closed mapping) preserve locally countable sn-networks,and a perfect preimage theorem on spaces with a locally countable sn-network is established.     

Tychonoff products of compact spaces in ZF and closed ultrafilters

Let {(X_i, T_i): i ∈I } be a family of compact spaces and let X be their Tychonoff product. ??(X) denotes the family of all basic non‐trivial closed subsets of X and ??_R(X) denotes the family of all closed subsets H = V × ΠX_i of X, where V is a non‐trivial closed subset of ΠX_i and Q_H is a finite non‐empty subset of I. We show: (i) Every filterbase ?? ? ??_R(X) extends to a ??_R(X)‐ultrafilter ? if and only if every family H ? ??(X) with the finite intersection property (fip for abbreviation) extends to a maximal ??(X) family F with the fip. (ii) The proposition “if every filterbase ?? ? ??_R(X) extends to a ??_R(X)‐ultrafilter ?, then X is compact” is not provable in ZF. (iii) The statement “for every family {(X_i, T_i): i ∈ I } of compact spaces, every filterbase ?? ? ??_R(Y), Y = Π_{i ∈I}Y_i, extends to a ??_R(Y)‐ultrafilter ?” is equivalent to Tychonoff's compactness theorem. (iv) The statement “for every family {(X_i, T_i): i ∈ ω } of compact spaces, every countable filterbase ?? ? ??_R(X), X = Π_{i ∈ω}X_i, extends to a ??_R(X)‐ultrafilter ?” is equivalent to Tychonoff's compactness theorem restricted to countable families. (v) The countable Axiom of Choice is equivalent to the proposition “for every family {(X_i, T_i): i ∈ ω } of compact topological spaces, every countable family ?? ? ??(X) with the fip extends to a maximal ??(X) family ? with the fip” (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Closed sets which are not -critical

In this paper we first observe that the complement of a countable closed subset of an -dimensional manifold has large -homology group. In the last section we use this information to prove that, under some topological conditions on the given manifold, certain families of fibers, in the total space of a fibration over , are not critical sets for some special real or -valued functions.

     

Complete theories with finitely many countable models. II

Previously, we obtained a syntactic characterization for the class of complete theories with finitely many pairwise non-isomorphic countable models [1]. The most essential part of that characterization extends to Ehrenfeucht theories (i.e., those having finitely many (but more than 1) pairwise non-isomorphic countable models). As the basic parameters defining a finite number of countable models, Rudin-Keisler quasiorders are treated as well as distribution functions defining the number of limit models for equivalence classes w.r.t. these quasiorders. Here, we argue to state that all possible parameters given in the characterization theorem in [1] are realizable. Also, we describe Rudin-Keisler quasiorders in arbitrary small theories. The construction of models of Ehrenfeucht theories with which we come up in the paper is based on using powerful digraphs which, along with powerful types in Ehrenfeucht theories, always locally exist in saturated models of these theories. Supported by RFBR grant Nos. 02-01-00258 and 05-01-00411. __________ Translated from Algebra i Logika, Vol. 45, No. 3, pp. 314–353, May–June, 2006.     

选择理论中两个结论的改进

本文利用可数序基的概念，讨论了下半连续集值映射的选择问题，改进了选择理论中的两个经典结论．     

L-模糊集的可数starplus-紧性

本文研究了在L-拓扑空间中,利用L-拓扑的水平拓扑引入可数starplus-紧性的概念,获得了可数starplus-紧性的性质,并且对一般拓扑中可数starplus-紧性的推广.     

星可数k网与序列覆盖s映射

证明了在空间具有星可数ｋ网的条件下,度量空间的１（２）序列覆盖ｓ映象是局部可分度量空间的１（２）序列覆盖、紧覆盖ｓ映象。