几乎弱加细空间

证明了如下结果:(1)空间X是几乎弱加细空间当且仅当X是几乎离散弱加细可膨胀的,并且X的每个开覆盖u={Uα:α∈Λ},都存在X的稠密子集D和u的开加细V=∪n∈ωVn,使得x∈D存在b∈ω和α∈Λ有x∈Uα,并且st(x,Vn)∪βα;(2)如果X=∏α∈λXα是|Λ|—仿紧空间,则X是几乎弱加     

Complexity of Categorical Theories with Computable Models

M. Lerman and J. Scmerl specified some sufficient conditions for computable models of countably categorical arithmetical theories to exist. More precisely, it was shown that if T is a countably categorical arithmetical theory, and the set of its sentences beginning with an existential quantifier and having at most n+1 alternations of quantifiers is _n+1 ⁰ for any n, then T has a computable model. J. Night improved this result by allowing certain uniformity and omitting the requirement that T is arithmetical. However, all of the known examples of theories of₀-categorical computable models had low level of algorithmic complexity, and whether there are theories that would satisfy the above conditions for sufficiently large n was unknown. This paper will include such examples.     

Two countably compact topological groups: One of size and the other of weight without non-trivial convergent sequences

E. K. van Douwen asked in 1980 whether the cardinality of a countably compact group must have uncountable cofinality in . He had shown that this was true under GCH. We answer his question in the negative. V. I. Malykhin and L. B. Shapiro showed in 1985 that under GCH the weight of a pseudocompact group without non-trivial convergent sequences cannot have countable cofinality and showed that there is a forcing model in which there exists a pseudocompact group without non-trivial convergent sequences whose weight is . We show that it is consistent that there exists a countably compact group without non-trivial convergent sequences whose weight is .

     

超实数域^＊R的拓扑结构

研究了超实数域上两种有用的拓扑－Ｑ－拓扑和Ｓ－拓扑。证明了以下结果：空间（＊Ｒ，Ｑ）是完全不连通的；＾＊Ｒ的Ｑ－紧子集只有有限集；＾＊Ｒ中的每一个银河是（＊Ｒ，Ｓ）的一个连通分支；＾＊Ｒ中的每一个具有有限长度的区间（不必是闭的）都是Ｓ－紧的，同时也纠正了《Ｍａｔｈ．Ｊａｐｏｎｉｃａ》上一篇论文中关于＾＊Ｒ上的Ｑ－拓扑的性质的一些错误。     

Shrinking property in Σ-products of paracompact p-spaces

It is shown that a Σ-product of paracompact p-spaces with countable tightness has the shrinking property.     

On box products

We prove two theorems about box products. The first theorem says that the box product of countable spaces is pseudonormal, i.e. any two disjoint closed sets one of which is countable can be separated by open sets. The second theorem says that assuming CH a certain uncountable box product is normal (i.e. <_ω1?□_α<ω1X_α where each Xα is a compact metric space).     

The Cohen--Lusk Conjecture

The question of Cohen and Lusk about the partial gluing of an orbit under a map of a free Z _p-space to ⁿ is answered in part.     

Nondiscrete P-groups can be reflexive

We present the first examples of nondiscrete reflexive P-groups (topological groups in which countable intersections of open sets are open) as well as of noncompact reflexive ω-bounded groups (precompact groups in which the closure of every countable set is compact). Our main result implies that every product of discrete Abelian groups equipped with the P-modified topology is reflexive. Taking uncountably many nontrivial factors, we thus answer a question posed by P. Nickolas and solve a problem raised by Ardanza-Trevijano, Chasco, Domínguez, and Tkachenko.New examples of non-reflexive P-groups are also given which are based on a further development of Leptin's technique going back to 1955.     

平行可分解格的若干性质

本文主要研究分配格中一个特殊子类,我们称它为平行可分解格,此类格有很多十分重要的特殊性质。