关于弱拟第一可数空间的注记

给出一个弱拟第一可数空间成为弱第一可数空间的充要条件,证明了空间是弱第一可数空间当且仅当它是具有序列点Gδ性质的弱拟第一可数空间且不含Sw的闭拷贝.同时还证明了每一弱第一可数空间(弱拟第一可数空间)都是某个第一可数空间的商二到一映像(商可数到一映像),作为应用,部分回答了林寿(2007)的一个问题.     

关于kr-空间与k-空间

本文给出了具有局部可数闭ｋ－网的ｋｒ－空间成为ｋ－空间的一个充要条件，此结果给林寿提出的问题作了一个回答。     

关于k-映射

本文给出度量空间ｋ－映象的一些新的特征。证明了ｋ－映射保持一些具有某种特定性质的点可数覆盖的空间。     

局部可数集族、局部有限集族与Alexandroff问题

本文引进分层强ｓ－映射和分层强紧映射建立具有σ－局部可数网、具有σ－局部可数ｋ－网、具有σ－局部可数基的正则空间以及σ－空间、－空间、ｇ－可度量空间和确定的度量空间之间的联系．这些都是对Ａｌｅｘａｎｄｒｏｆｆ问题的回答．     

局部可数集族、局部有限集族与Alexandroff问题

本文引进分层强ｓ－映射和分层强紧映射建立具有σ－局部可数网、具有σ－局部可数ｋ－网、具有σ－局部可数基的正则空间以及σ－空间、－空间、ｇ－可度量空间和确定的度量空间之间的联系．这些都是对Ａｌｅｘａｎｄｒｏｆｆ问题的回答．     

csf可数空间的注记

讨论csf可数空间的性质,把csf可数空间刻画为度量空间的映像,同时探讨了伪紧的csf可数空间的第一可数性质,推广了Arhangel’skiˇ?关于度量空间伪开s映像的结果,证明了正则伪紧的仿拓扑群是可度量化的当且仅当它是csf可数的Fr′echet空间.     

关于Lanev空间

本文给出非-空间的Lasnev空间的一个特征。作为它的推论,我们证明了一个Lasnev空间是一个?-空间当且仅当它具有点可数cs-网。     

关于Lanev空间

本文给出非-空间的Lasnev空间的一个特征。作为它的推论,我们证明了一个Lasnev空间是一个?-空间当且仅当它具有点可数cs-网。     

Michael-Nagami问题的注记

本文证明Hausdorff空间Y是度量空间的强紧覆盖商s映象当且仅当Y是具有点可数cs网络的序列空间,它部分地回答了Michael-Nagami问题.     

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。     

On the Decomposition of Order-Separable Posets of Countable Width into Chains

Order - A partially ordered set X has countable width if and only if every collection of pairwise incomparable elements of X is countable. It is order-separable if and only if there is a countable...     

Spaces in Which Countable Closed Sets Have Countable Character

It is proved that in a T ₃ space countable closed sets have countable character if and only if the set of limit point of the space is a countable compact set and every compact set is of countable character. Also, it is shown that spaces where countable sets have countable character are WN-spaces and are very close to M-spaces. Finally, some questions of Dai and Lia are discussed and some questions are proposed.     

Modules with Few Types over a Commutative Valuation Ring

It is proved that a module M over a countable commutative valuation ring has few models if and only if M is ω-stable. Let T be an arbitrary complete theory of modules over either a countable serial ring or a countable commutative Prüfer ring or a countable hereditary Noetherian prime ring. It is proved that Martin's conjecture is true for T. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 74, Algebra-15, 2000.     

度量空间的序列复盖SS一映射

证明了如下结果：（l）拓扑空间X具有局评可数弱基当且仅当X＃A星空间的1一序列复盖商ss-掩映象；（2）拓升空间X具有局都可数基当且仅当XRk量空间的2一序列复盖商ss一映象．     

Complete congruence lattices of two related modular lattices

A topological monoid is isomorphic to an endomorphism monoid of a countable structure if and only if it is separable and has a compatible complete ultrametric such that composition from the left is non-expansive. We also give a topological characterisation of those topological monoids that are isomorphic to endomorphism monoids of countable \({\omega}\)-categorical structures. Finally, we present analogous characterisations for polymorphism clones of countable structures and for polymorphism clones of countable \({\omega}\)-categorical structures.     

On the Countability of Noetherian Dimension of Modules

Abstract

It is shown that a module M has countable Noetherian dimension if and only if the lengths of ascending chains of submodules of M has a countable upper bound. This shows in particular that every submodule of a module with countable Noetherian dimension is countably generated. It is proved that modules with Noetherian dimension over locally Noetherian rings have countable Noetherian dimension. We also observe that ω^ω is a universal upper bound for the lengths of all chains in Artinian modules over commutative rings.     

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)     

Algebraic Characterization of Dense-Separability Among Compact Spaces

In this note, we show that a compact Hausdorff space X is dense-separable if and only if every family of ideals of C(X) with zero intersection has a countable subfamily with zero intersection. As a consequence of this characterization we observe that every compact dense-separable space whith Soc(C(X)) = 0 has a countable dense and co-dense subset.     

关于CSS空间及相关结论

CSS空间是指空间中的紧集都是一致G_δ集的空间.该文的第一部分,主要证明了具有拟G_δ(2)对角线的空间是CSS空间.另外,还证明了如果X是可数个闭的CSS空间的并,则X是CSS空间.CSS空间的可数积空间是CSS空间;第二部分证明了如果空间X可以表示成可数个闭的β空间(或半层空间)的并,则X是β空间(或半层空间).     

Classification of cocycles of generic equivalence relations

We study cocycles of an ergodic generic countable equivalence relation ℜ modulo meager sets. Two cocycles of ℜ are called weakly equivalent if they are cohomologous up to an element of Aut ℜ. It is proved that two nontransient cocycles with values in an arbitrary countable group are weakly equivalent if and only if their generic Mackey actions are isomorphic.