点是Gδ-集的∑*-空间的构成定理

在林寿与我最近合作的一篇文章中指出了∑*-空间的构成定理需重新考虑.本文就是要证明在空间X的每个点是Gδ-集的条件下该构成定理是成立的,所得的结论是:X是T1且每个点是Gδ-集的∑*-空间,如果f:X→Y是闭的满连续映射,则在Y中有一σ-闭离散子空间Z,使得对每个y∈Y\Z,f-1(y)是X的w1-紧子空间.为得到该主要结果,本文证明了若空间X是每个点是Gδ-集的次亚紧空间.则X中的每个闭离散子集是X中的Gδ-集.     

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.     

点是Gδ^－集的∑^＊－空间的构成定理

在林寿与我最近合作的一篇文章中指出了∑^*-空间的构成定理需重新考虑．本文就是要证明在空间X的每个点是Gδ^-集的条件下该构成定理是成立的，所得的结论是：X是T1且每个点是Gδ^-集的∑^ -空间，如果f：X→Y是闭的满连续映射，则在Y中有-σ-闭离散子空间Z，使得对每个y∈Y＼Z，f^-1(y)是X的ω1^-紧子空间．为得到该主要结果，本文证明了若空间X是每个点是Gδ^-集的次亚紧空间．则X中的每个闭离散子集是X中的Gδ^-集．     

拓扑群中广义度量性质的一个注记

主要讨论拓扑群中的一些广义度量性质.证明了对于拓扑群G和G的局部紧度量子群H,如果商群G/H是层空间(半层空间,κ半层空间,σ空间),则G也是层空间(半层空间, κ半层空间,σ空间),这肯定回答了Arhangel'skii A.V.和Uspenskij V.V.提出的一个问题.同时还讨论了弱拟第一可数的,不含Sω的闭拷贝的仿拓扑群.     

空间的-特殊点可数族使其为D-空间

本文给出了空间为D-空间的-充分条件,主要结论如下:如果空间X有一点可数族F,满足对X的任-子集A (C) X,若A在X中不闭,都存在某点x ∈-A\A,使得对X的任一开集U,若X∈U,都存在某个F∈F,使得X ∈F(C) U且F ∩ A≠ (θ),则X是D-空间.由此结论,我们得到-序列空间若有点可数cs*-网络,则X是D-空间.     

强S-Lindel(o)f空间

利用强半开集定义了强S-Lindel(o)f可数性的概念,讨论了其基本性质.证明了强S-Lindel(o)f性对强半闭子集是遗传的,具有强半同胚不变性等,并给出了强S-Lindel(o)f空间的一些应用.     

空间的一特殊点可数族使其为D-空间

本文给出了空间为D-空间的一充分条件，主要结论如下：如果空间X有一点可数族厂，满足对X的任一子集A X，若A在X中不闭，都存在某点X∈A＼A，使得对X的任一开集U，若X∈U，都存在某个F∈F，使得X∈F U且F∩A≠D，则X是D-空间．由此结论，我们得到一序列空间若有点可数C8^8-网络，则X是D-空间．     

Lukasiewicz语义集上的紧Hausdorff拓扑

以Ω记从全体命题之集F（S）到单位区间的全体Lukasiewicz赋值之集.本文通过一种自然的方法在Ω上引入了Fuzzy拓扑δ,证明了其为第二可数的零维良紧空间,并证明了δ在Ω上生成的截拓扑空间是第二可数的紧Hausdorff空间,从而是可度量化的空间.     

单调空间与度量化定理

本文回答了关于MCM空间遗传性的一个问题,讨论了k-MCM空间是k半层空间的条件,得到了一些用g函数刻划的度量化定理.主要结论有:MCM空间是关于Fσ子空间遗传的;在正规空间类中,q空间(ωN空间,k-MCM空间)是关于开Fσ子空间遗传的;如果X是具有Gδ对角线的正则次中紧 k-MCM空间,则X是k半层空间;X是可度量化空间的充要条件是存在X上的g函数满足对X中任意不相交的闭集F与紧集C,都有某个n∈ω,使得(∪x∈F g(n,x))∩(∪y∈C g(n,y))=(?).     

边缘s映射、边缘紧映射与k半层空间

主要讨论了k半层空间上的闭映射性质,证明了k半层空间的闭映像若是不含有闭子空间同胚于S_(ω1)(S_ω)的k空间,则该闭映射是边缘s映射(边缘紧映射).最后给出例子表明弱层空间未必是层空间,否定回答了关于层空间的一个问题.     

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.     

The D-property in unions of scattered spaces

We show in a direct way that a space is D if it is a finite union of subparacompact scattered spaces. This result cannot be extended to countable unions, since it is known that there is a regular space which is a countable union of paracompact scattered spaces and which is not D. Nevertheless, we show that every space which is the union of countably many regular Lindelöf C-scattered spaces has the D-property. Also, we prove that a space is D if it is a locally finite union of regular Lindelöf C-scattered spaces.     

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.     

Sacks forcing and the total failure of Martin's Axiom

Using side-by-side Sacks forcing, it is proved relatively consistent that the continuum is large and Martin's Axiom fails totally, that is, every c.c.c. space is the union of ?₁ nowhere dense sets (equivelently, if P is a nontrivial partial ordering with the countable chain condition, then there are ?₁ dense sets in P such that no filter in P meets them all).     

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.     

Topological vector spaces, compacta, and unions of subspaces

In the first two sections, we study when a σ-compact space can be covered by a point-finite family of compacta. The main result in this direction concerns topological vector spaces. Theorem 2.4 implies that if such a space L admits a countable point-finite cover by compacta, then L has a countable network. It follows that if f is a continuous mapping of a σ-compact locally compact space X onto a topological vector space L, and fibers of f are compact, then L is a σ-compact space with a countable network (Theorem 2.10). Therefore, certain σ-compact topological vector spaces do not have a stronger σ-compact locally compact topology.In the last, third section, we establish a result going in the orthogonal direction: if a compact Hausdorff space X is the union of two subspaces which are homeomorphic to topological vector spaces, then X is metrizable (Corollary 3.2).     

When a compact (countably compact) set is closed. II

We obtain (a) necessary and sufficient conditions and (b) sufficient conditions for a compact (countably compact) set to be closed in products (sequential products) and subspaces (sequential subspaces) of normal spaces. As a consequence of these, sufficient conditions are obtained for (i) the closedness of arbitrary (countable) union of closed sets and (ii) the equality of the union of the closures and the closure of the union of arbitrary (countable) families of sets in these spaces. It is also shown that these results do not hold for quotients of even T ₄,-spaces.     

Cosmic spaces which are not μ-spaces among function spaces with the topology of pointwise convergence

A space is called a μ-space if it can be embedded in a countable product of paracompact F_σ-metrizable spaces. The following are shown:(1) For a Tychonoff space X, if C_p(X,R) is a μ-space, then X is a countable union of compact metrizable subspaces.(2) For a zero-dimensional space X, C_p(X,2) is a μ-space if and only if X is a countable union of compact metrizable subspaces.In particular, let P be the space of irrational numbers. Then C_p(P,2) is a cosmic space (i.e., a space with a countable network) which is not a μ-space.     

SMOOTHNESS PROPERTIES OF REAL-VALUED FUNCTIONS BASED ON THEIR OSCILLATION

Abstract

We consider the following two selection principles for topological spaces:

Principle 1: For each sequence of dense subsets, there is a sequence of points from the space, the n-th point coming from the n-th dense set, such that this set of points is dense in the space;

Principle 2: For each sequence of dense subsets, there is a sequence of finite sets, the n-th a subset of the n-th dense set, such that the union of these finite sets is dense in the space.

We show that for separable metric space X one of these principles holds for the space C_p (X) of realvalued continuous functions equipped with the pointwise convergence topology if, and only if, a corresponding principle holds for a special family of open covers of X. An example is given to show that these equivalences do not hold in general for Tychonoff spaces. It is further shown that these two principles give characterizations for two popular cardinal numbers, and that these two principles are intimately related to an infinite game that was studied by Berner and Juhász.     

cut* 空间的拓扑性质

该文引入了 cut^*空间的概念,所谓的 cut^*空间是指去掉任意一点连通,去掉任意两点不连通的连通空间.通过对其性质的讨论,得到如下主要结论: 首先得到cut^*空间中每个点非开即闭,并且cut^*空间中有无限多个闭点;其次讨论了一类特殊的 cut^*空间,即去掉一点是COTS的 cut^* 空间.指出``$X$是 cut^*空间,任意 $x\inX,X\setminus\{x\}$是不可约cut空间'这样的空间类是不存在的.在文章的最后,讨论了去掉一点是LOTS的 cut^*空间的覆盖性质,得到这样的空间是紧空间或Lindel\"of空间的结论.