关于仿紧性与拓扑空间的可度量性

引入σ-相对垫状开加细等概念并用它们刻划了仿紧性.引入线性遗传闭包保持集族的概念;证明了下列度量化定理:正则T_1空间是可度量的当且仅当它具有可数伪特征且有σ-线性遗传闭包保持基.     

Fuzzy拓扑空间中的紧性(Ⅰ)——定义和特征(英文)

本文在引入了一复盖的概念之后,定义了(?)一紧性,得出了关于闭集中心族,F-网与F-滤子的(?)-紧性的特微,以及A1exander子基定理。并进一步定义了S-紧,L-紧,I-紧和F-紧性,讨论了这些概念之间的关系。设A,B∈I~Y为X中的Fuzzy集,我们称有序对〈A,B〉为X中的一个(?)一集。定义1 设(X,F)是一个Fuzzy拓扑空间,〈A,B〉为X中的一个(?)一开集,P∈P_*(X)。如果〈A,B〉是P的邻域,则我们说〈A,B〉覆盖P。一个开(?)一集族(?)={〈A_λ,B_λ〉:λ∈Λ}称为X的一个(?)-覆盖,当且仅当对于任一P∈IP_*(X),存在λ∈Λ,使〈A_λ,B_λ>覆盖P。定义2 Fuzzy拓扑空间(X,F)称为(?)-紧的,当且仅当每个(?)覆盖都有有限子(?)-覆盖。定理1 Fuzzy拓扑空间(X,F)是(?)-紧的,当且仅当每个闭(?)-集构成的有限中心族都是中心族。定理2 Fuzzy拓扑空间(X,F)是(?)-紧的,当且仅当X中的每个F-网或者(?)-滤子都有聚点。定理5 设S为Fuzzy拓扑空间(X,F)的一个子基,若每个(?)覆盖(?)={〈A_λ,B_λ〉:A_λ,B_λ∈S,λ∈Λ}都有有限子覆盖,则(X,F)是(?)-紧的。     

正则半开集和半同胚空间类

设A■X,v为X上的拓扑,用A_v~0、A_v~-、(A_v)_0、(A_v)_-、A’_v分别记A在(X,v)中的内部、闭包、半内部、半闭包和补集。用η_v、ζ_v、ζ_v分别记(X,v)中的半开集族、无处稠密集族和稠密集族。当v=u时,略去上述记号中的足码v。令[u]={v|v为X上的拓扑且     

定向闭包空间的若干性质

拓扑空间X称为定向闭包空间(简称DC空间),若它是T₀的，且其任一既约闭集都是某定向子集的闭包，此处X赋予特殊化序。本文讨论了定向闭包空间的一些基本性质，证明了偏序集赋予Alexandroof拓扑所得空间和其Smyth幂空间都是DC空间；偏序集赋予上拓扑是quasisober空间当且仅当它是DC空间；DC性对开子空间遗传，但对饱和子空间一般不遗传；对T₀空间X,其Smyth幂空间是DC空间一般不蕴含X是DC空间。     

关于闭包保持和定理

<正> R.E.Hodel在总结前人成果的基础上首先给出了拓扑空间的局部有限和的一般性定理.M.K.Singal及S.P.Arya改进了上述结果,给出了更一般性的局部有限和定理.Singal-Arya就遗传性闭包保持和的情况给出了更一般性的和定理.本文也就遗传性闭包保持和的情况给出更一般性的和定理(定理1、2)以改进[4]中的相应结果,并给出了使可数遗传性闭包保持、局部可数闭和定理成立的一般性条件(定理3).     

具有σ遗传闭包保持双网的空间

证明了正则空间X具有σ遗传闭包保持双网当且仅当或X是cosmic空间，或X是σ闭离散空间，回答了[1]提出的一个问题．     

L_Δ-闭包空间的Δ-仿紧性

在L_Δ-闭包空间中借助于L_α-Δ-开覆盖、α-局部有限集族、α--局部有限集族引入了Δ-仿紧性.证明了它对Δ-闭集是遗传的、Δ-紧的L_Δ-闭包空间和Δ-仿紧空间的积空间是Δ-仿紧的重要性质.     

半同胚空间类中存在最弱拓扑的条件

若(X,τ)是 S_1-空间,S_τ是它的半开集族[τ]={σ:σ为 X 的拓扑且 S_σ=S_τ)。本文到如下结果:1)若[τ]有最弱拓扑τ(?),则(X,τ(?))是(X,τ)的半正则化空间。2)[τ]中有最弱拓扑的充要条件是(X,τ)的每个非空开集都包含非空的正则开集。因为 T_1一空间是 S_1空间,伪度量空间是 S_1一空间但未必是 T_1一空间。所以,我们的结果推广了[1]中的定理5、推论5和定理6。     

半同胚空间类

集S(?)X称为半开的,如果存在开集U使得U(?)S(?)U~-,这里()-表示闭包算子。本文讨论了和(X,u)具有相同半开集族的X上的全体拓扑所构成的族(u)的结构。首先,给出了[u]中拓扑的一般形式和[u]中存在最弱拓扑的充要条件。其次,指出了若干非半拓扑性质,最后,讨论在拓扑空间的各种运算下,半同胚空间类的变化情况。     

■_0-空间及其推广空间的距离化

在〔1〕中,E.Michael 引入了(?)_0-空间的概念,本文给出(?)_0-空间的几个度量化定理,推广了 Michael 的一些结果,并将(?)_0-空间在函数空间的性质也作了进一步的推广.拓扑空间 X 的子集族(?)是 X 的伪基,对 X 中任一紧子集 K 和开集 U,若 K(?)U,则(?)B∈(?),使得 K(?)B(?)U。正则的且有可数伪基的拓扑空间叫(?)_0-空间。     

M1-空间的和定理

In this gaper, two sum theorems for M₁-spaces are obtained. These results improve two results obtained by Gao Guoshi in [ 2 ] respectively. A result for M₀-spaces is also obtained which answers J.Nagata's question partly.     

Hereditarily strongly cwH and other separation axioms

We explore the relation between two general kinds of separation properties. The first kind, which includes the classical separation properties of regularity and normality, has to do with expanding two disjoint closed sets, or dense subsets of each, to disjoint open sets. The second kind has to do with expanding discrete collections of points, or full-cardinality subcollections thereof, to disjoint or discrete collections of open sets. The properties of being collectionwise Hausdorff (cwH), of being strongly cwH, and of being wD(ℵ₁), fall into the second category. We study the effect on other separation properties if these properties are assumed to hold hereditarily. In the case of scattered spaces, we show that (a) the hereditarily cwH ones are α-normal and (b) a regular one is hereditarily strongly cwH iff it is hereditarily cwH and hereditarily β-normal. Examples are given in ZFC of (1) hereditarily strongly cwH spaces which fail to be regular, including one that also fails to be α-normal; (2) hereditarily strongly cwH regular spaces which fail to be normal and even, in one case, to be β-normal; (3) hereditarily cwH spaces which fail to be α-normal. We characterize those regular spaces X such that X×(ω+1) is hereditarily strongly cwH and, as a corollary, obtain a consistent example of a locally compact, first countable, hereditarily strongly cwH, non-normal space. The ZFC-independence of several statements involving the hereditarily wD(ℵ₁) property is established. In particular, several purely topological statements involving this property are shown to be equivalent to b=ω₁.     

On some spaces which can be partitioned by the rational line

Extending the Main Theorem of [8], we consider in this paper some topological spaces (not necessarily metrizable) which can be partitioned by the rational line. Based on W. Sierpiski's theorem (Lemma 1) the following results are obtained. Theorem. A regular, first countable, self-dense and hereditarily paracompact space can be partitioned by the rational line. Theorem. A hereditarily paracompact orderable space can be partitioned by the rational line iff it is self-dense and first countable. Some important non-metrizable spaces satisfying the hypotheses of the above theorems, as examples, are enumerated. We take orderable _µ-metric spaces (u>0) as example to illustrate the fact: there exist spaces which are hereditarily paracompact, orderable but can't be partitioned by the rational line.     

Кусочно-монотонная и рациональная аппрок симации и равномерная сходим ость рядов Фурье

Let $$R_n [f] = \inf \left\{ {\mathop {\max }\limits_{ - \pi \leqq x \leqq \pi } \left| {f(x) - \frac{{P(x)}}{{Q(x)}}} \right|} \right\}$$ , whereP andQ denote polynomials (algebraic or trigonometric) of degree ≦n. Theorem 2a. If for a continuous 2π-periodic function f the condition $$\sum\limits_{n = 1}^\infty {\frac{1}{n}R_n [f]< \infty } $$ holds, then the Fourier series of f converges to f(x) uniformly. Theorem 2b. Let {β _n } be a non-increasing sequence of positive numbers such that $$\sum\limits_{n = 1}^\infty {\frac{1}{n}\beta _n = \infty } $$ Then there exists a continuous 2π-periodic function f₀ for which R_n[f₀]≦β_n for all n=1 and yet the Fourier series of f₀ diverges at x=0.R _n [f]may be replaced in these theorems byM _n [f], whereM _n [f] is the minimal uniform deviation off(x) from piecewise monotonie functionsМ _n (х) of order ≦n.     

Characterization of 3 D 4(q)

The author defined the concept order components in [2] and gave a new characterization of sporadic simple groups by their order components in [7]. Afterwards the following groups were characterized by the author: G₂(q), q = 0 (mod 3)^[8]; E₈(q)^[9]; Suzuki-Ree groups^[10]; PSL₂(q)^[11]. Here the author will continue such kind of characterization and prove that:Theorem 1. Let G be a finite group, M = ³D₄(q). If G and M has the same order components, then G M.And the following theorems follows from Theorem 1.Theorem 2. (Thompsons Conjecture) Let G be a finite group, Z(G) = 1,M = ³D₄(q). If N(G) = N(M), then G M. (ref. [6])Theorem 3. (Wujie Shi) Let G be a finite group, M = ³D₄(q). If|G| = |M|, _e(G) = _e(M), then G M. (ref. [15])All notations are the same as in [2]. According to the classification theorem of finite simple groups, [12] and [13], we can list the order components of finite simple groups with nonconnected prime graphs in Tables 1-4 (ref. [5]).American Mathematics Society Classification 20D05 20D60The author is indebted to Fred and Barbara Kort Sino-Israel Postdoctoral Programme for supporting my post-doctoral position (1999.10-2000.10) at Bar-Ilan University, also to Emmy Noether Mathematics Institute and NSFC for partially financial support.     

The Baer-invariant of a semidirect product

In 1972 K.I. Tahara [7,2, Theorem 2.2.5], using cohomological methods, showed that if a finite group is the semidirect product of a normal subgroup N and a subgroup T, then M(T) is a direct factor of M(G), where M(G) is the Schur-multiplicator of G and in the finite case, is the second cohomology group of G. In 1977 W. Haebich [1, Theorem 1.7] gave another proof using a different method for an arbitrary group G.In this paper we generalize the above theorem. We will show that scN_cM(T) is a direct factor of _cM(G), where _c[3, p. 102] is the variety of nilpotent groups of class at most c ≥ 1 and _cM(G) is the Baer-invariant of the group G with respect to the variety _c [3, p. 107].     

On <Emphasis Type="Italic">k</Emphasis>-Spaces and <Emphasis Type="Italic">k</Emphasis><Subscript><Emphasis Type="Italic">R</Emphasis></Subscript>-Spaces

In this note we study the relation between k _R -spaces and k-spaces and prove that a k _R -space with a σ-hereditarily closure-preserving k-network consisting of compact subsets is a k-space, and that a k _R -space with a point-countable k-network consisting of compact subsets need not be a k-space. This work was supported by the NSF of China (10271056).     

A note on the categorical van Kampen Theorem

In this note we show that for the canonical map X₁+X₂→X, from the topological sum of two subspaces X₁, X₂ of X into X, to be of effective descent it is sufficient to be hereditarily a quotient map. Further, making use of the Brown-Janelidze categorical van Kampen Theorem, this translates into a van Kampen-type result.     

Exploring hypergraphs with martingales

Recently, in [Random Struct Algorithm 41 (2012), 441–450] we adapted exploration and martingale arguments of Nachmias and Peres [ALEA Lat Am J Probab Math Stat 3 (2007), 133–142], in turn based on ideas of Martin‐Löf [J Appl Probab 23 (1986), 265–282], Karp [Random Struct Alg 1 (1990), 73–93] and Aldous [Ann Probab 25 (1997), 812–854], to prove asymptotic normality of the number L₁ of vertices in the largest component of the random r‐uniform hypergraph in the supercritical regime. In this paper we take these arguments further to prove two new results: strong tail bounds on the distribution of L₁, and joint asymptotic normality of L₁ and the number M₁ of edges of in the sparsely supercritical case. These results are used in [Combin Probab Comput 25 (2016), 21–75], where we enumerate sparsely connected hypergraphs asymptotically. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 325–352, 2017     

On resolutions of generalized metric spaces

We study the relation between generalized metric spaces and their resolutions. Here, we consider the class of spaces with G_δ-diagonals, M₃-,M₁-spaces, M₃–μ-spaces and developable spaces.