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．     

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.     

函数空间上的Cauchy收敛拓扑(英文)

本文研究了度量空间X到实直线R上的连续函数空间C(X,R)上的Cauchy收敛拓扑Tc.u,点态收敛拓扑Tp.u,紧开拓扑Tk和一致收敛拓扑Tu相等的等价条件.利用Cauchy覆盖得到了(C(X,R),Tc.u)的特征与X的Cauchy覆盖数相等的一个对偶定理,获得了(C(X,R),Tc.u)可度量化当且仅当(C(X,R),Tc.u)是第一可数的当且仅当X具有可数Cauchy覆盖数,肯定地回答了Michael H Clapp等在文献[1]中提到的问题.     

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).     

Lindelöf type of generalization of separability in Banach spaces

We will introduce the countable separation property (CSP) of Banach spaces X, which is defined as follows: X has CSP if each family E of closed linear subspaces of X whose intersection is the zero space contains a countable subfamily E₀ with the same intersection. All separable Banach spaces have CSP and plenty of examples of non-separable CSP spaces are provided. Connections of CSP with Marku?evi?-bases, Corson property and related geometric issues are discussed.     

Division and k-th root theorems for Q-manifolds

We prove that a locally compact ANR-space X is a Q-manifold if and only if it has the Disjoint Disk Property (DDP), all points of X are homological Z∞-points and X has the countable-dimensional approximation property (cd-AP), which means that each map f:K→X of a compact polyhedron can be approximated by a map with the countable-dimensional image. As an application we prove that a space X with DDP and cd-AP is a Q-manifold if some finite power of X is a Q-manifold. If some finite power of a space X with cd-AP is a Q-manifold, then X2 and X×[0,1] are Q-manifolds as well. We construct a countable familyχof spaces with DDP and cd-AP such that no space X∈χis homeomorphic to the Hilbert cube Q whereas the product X×Y of any different spaces X, Y∈χis homeomorphic to Q. We also show that no uncountable familyχwith such properties exists.     

Semipolar Sets and Intrinsic Hausdorff Measure

Potential Analysis - Given a “Green function” G on a locally compact space X with countable base, a Borel set A in X is called G-semipolar, if there is no measure ν ≠ 0...     

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。     

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.     

具有可数弱基空间的一个新特征

本文证明了正则空间Ｘ具有可数弱基当且仅当它是具有可数ｋ网的ｋ空间并且不含闭子空间同胚于Ｓｘ。它肯定地回答了Ｙ．Ｔａｎａｋａ在１９８３年提出的一个问题。     

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.     

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.     

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.     

Spaces with a Souslin and a shanin condition

If X is a regular hereditary Souslin space and x ∈X then either there exists a sequence {x_n: n=1, 2, ...} ? X{x} such that x ∈ [{x_n∶n=1, 2, ...}], or the pseudocharacter of x in X is no greater than countable. In other words, if X is a hereditary Souslin bicompactum which is a χ-space, then X is a Frechet-Urysohn space.     

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.     

Note on the Construction of Free Monoids

We construct free monoids in a monoidal category with finite limits and countable colimits, in which tensoring on either side preserves reflexive coequalizers and colimits of countable chains. In particular this will be the case if tensoring preserves sifted colimits.     

$C(X)$上的开点与紧开拓扑

In this note we define a new topology on C(X),the set of all real-valued continuous functions on a Tychonoff space X.The new topology on C(X) is the topology having subbase open sets of both kinds:[f,C,ε[={g E C(X):|f(x)-g(x)| ε for every x∈C} and[U,r]~-={g∈C(X):g~(-1)(r)∩U≠φ},where f∈C(X),C∈KC(X)={nonempty compact subsets of X},ε 0,while U is an open subset of X and r∈R.The space C(X) equipped with the new topology T_(kh) which is stated above is denoted by C_(kh)(X).Denote X_0={x∈X:x is an isolated point of X} and X_c={x∈X:x has a compact neighborhood in X}.We show that if X is a Tychonoff space such that X_0=X_c,then the following statements are equivalent:(1) X_0 is G_δ-dense in X;(2) C_(kh)(X) is regular;(3) C_(kh)(X) is Tychonoff;(4) C_(kh)(X) is a topological group.We also show that if X is a Tychonoff space such that X_0=X_c and C_(kh)(X) is regular space with countable pseudocharacter,then X is σ-compact.If X is a metrizable hemicompact countable space,then C_(kh)(X) is first countable.     

Cc （X） Spaces with X Locally Compact

In this paper, we show, among other results, that if X is a [separable] locally compact space X [satisfying the first countability axiom] then the space Cc （X） has countable tightness [if and only if it has bounding tightness] if and only if it is Frechet-Urysohn, if and only if Cc （X） contains a dense （LM） subspace and if and only if X is a-compact.     

On the Countable Tightness of Product Spaces

In this paper, we discuss the countable tightness of products of spaces which are quotient simages of locally separable metric spaces, or k-spaces with a star-countable k-network. The main result is that the following conditions are equivalent： （1） b = ω1; （2） t（Sω×Sω1） 〉 ω; （3） For any pair （X, Y）, which are k-spaces with a point-countable k-network consisting of cosmic subspaces, t（X×Y）≤ω if and only if one of X, Y is first countable or both X, Y are locally cosmic spaces. Many results on the k-space property of products of spaces with certain k-networks could be deduced from the above theorem.     

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)