Extremal k-pseudocompact abelian groups
Anna Giordano Bruno
对基数k，推广Comfort和van Mill的一个结果，证明了每个权>k的k-伪紧阿贝尔群都有一定的真稠密k-伪紧子群，并容许一定的严格精细k-伪紧群拓扑.
For a cardinal k, generalizing a recent result of Comfort and van Mill, we prove that every k-pseudocompact abelian group of weight >k has some proper dense k-pseudocompact subgroup and admits some strictly finer k-pseudocompact group topology.
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A space X is selectively sequentially pseudocompact if for every sequence (U_n) of non-empty open subsets of X, one can choose a point x_n in each U_n in such a way that the sequence (x_n) has a convergent subsequence. Let G be a group from one of the following three classes: (i) V-free groups, where V is an arbitrary variety of Abelian groups; (ii) torsion Abelian groups; (iii) torsion-free Abelian groups. Under the Singular Cardinal Hypothesis SCH, we prove that if G admits a pseudocompact group topology, then it can also be equipped with a selectively sequentially pseudocompact group topolo…
On the quasi-component of pseudocompact abelian groups
本文讨论了（完美）极小伪紧阿贝尔群G的拟分支q（G）与其完备化的拟分支q（\widetilde G）之间的关系.具体地说，我们刻画了紧连通阿贝尔群C和子群A的对偶（C，A）使得\cong q（G）和\cong q（\widetilde G）.结果表明，对每个正整数n或n=ω，存在大量的阿贝尔伪紧完美极小n维群G，使得G的拟分支在G的完备化的拟分支中不稠密.
Pseudocompact group topologies with no infinite compact subsets
我们证明了每个满足温和基数不等式的阿贝尔群都有一个伪紧群拓扑，所有可数子群都从这个伪紧群拓扑继承极大完全有界拓扑（我们说这样的拓扑满足性质$\h$）。每个具有基数$|G|\leq 2^{2^\cc}$的伪紧阿贝尔群$G$都满足这个不等式，因此允许具有性质$\h$的伪紧群拓扑。在奇异基数假设（SCH）下，这个判据可以与伪紧群的代数结构分析相结合，证明每个伪紧阿贝尔群都有一个伪紧群拓扑具有性质$\h$ .我们还观察到，伪紧阿贝尔群的性质$\h$不包含无限紧子集，庞特里亚金自反预紧群的例子是不紧的。
Minimal pseudocompact group topologies on free abelian groups
一个Hausdorff拓扑群G是极小的，如果G与一个Hausdorff拓扑群H之间的每个连续同构f：G --> H是开的.证明了如下定理：对任意无穷小阿贝尔群G，存在一个基数序列{σ_n：nPDMATHABMM N}，使得w（G）= sup {σ_n：nPDMATHABMM N}，且sup {2^{σ_n}：nPDMATHABMM N} \leq |G| \leq 2^{w（G）}，其中w（G）是G的权.若G是一个无穷极小阿贝尔群，则对于某个基数σ，|G| = 2^σ，或w（G）= min {σ：|G| \leq 2^σ};而且，等式|G| = 2^{w（G）}在cf（w（G））> ω时成立。对于基数κ，我们用F_κ表示具有κ个生成元的自由阿尔贝群.若F_κ容许伪紧群拓扑，则有κ \geq c，其中c是连续统的基数.证明了F_c上极小伪紧群拓扑的存在性等价于Lusin的假设2^{ω_1} = c.当κ> c时，证明了F κ允许（零维）极小伪紧群拓扑当且仅当F κ既有极小群拓扑又有伪紧群拓扑.若κ> c，则F_κ允许权为σ的连通极小伪紧群拓扑当且仅当κ= 2^σ.最后，我们建立了没有无限无挠阿贝尔群可以配备一个局部连通极小群拓扑。
Precompact noncompact reflexive abelian groups
给出了一系列预紧、非紧、自反的拓扑阿贝尔群的例子。其中有些群是伪紧的，甚至是可数紧的，但我们证明了也存在预紧的非伪紧自反群。证明了每个伪紧阿贝尔群都是自反伪紧群关于闭自反伪紧子群的商。
We say that a topological space X is selectively sequentially pseudocompact (SSP for short) if for every sequence (U_n) of non-empty open subsets of X, one can choose a point x_n in U_n for every n in such a way that the sequence (x_n) has a convergent subsequence. We show that the class of SSP spaces is closed under taking arbitrary products and continuous images, contains the class of all dyadic spaces and forms a proper subclass of the class of strongly pseudocompact spaces introduced recently by García-Ferreira and Ortiz-Castillo. We investigate basic properties of this new class and its r…
Compact-like abelian groups without non-trivial quasi-convex null sequences
本文研究了不含序列{x_n}的预紧阿贝尔群G，使得{0} \cup { \pm x_n：n \in N}在G中是无穷的拟凸的，且x_n --> 0.我们在下列群类中刻画了具有此性质的群：（a）有界预紧阿贝尔群;（b）极小阿贝尔群;（c）完全极小阿贝尔群;（d）ω-有界阿贝尔群。给出了具有此性质的极小阿贝尔群的例子，并证明了存在具有此性质的极小伪紧阿贝尔群;此外，在Martin公理下，此群可被选为可数紧极小阿贝尔群.
Pontryagin duality in the class of precompact Abelian groups and the Baire property
我们提出了一个广泛的自反，预紧，非紧，阿贝尔拓扑群$G$确定的三个要求。它们必须具有Baire性质，满足\textit{open refinement condition}，并且不包含无限紧子集。这种性质的组合保证了对偶群$G^\wedge$的所有紧致子集都是有限的。我们还证明了许多（非自反）预紧阿贝尔群是自反预紧阿贝尔群的等价群。这包括所有具有Baire性质的准紧几乎度量化群及其乘积。最后，给定一个紧阿贝尔群$G$，其权为$\geq 2^\om$，我们找到了$G$的适当稠密子群$H_1$和$H_2$，使得$H_1$是自反的伪紧的，而$H_2$是非自反的几乎可度量化的.
Precompact abelian groups and topological annihilators
对于紧Hausdorff阿贝尔群K及其子群H，定义H在K中的g-闭包g（H）为由$χ\in K$组成的子群，使得对于H在$\hat K$上诱导的拓扑中收敛到0的$\hat K$（K的庞特里亚金对偶）中的每个序列{a_n}，T中的$χ(a_n)\longrightarrow 0$ =R/Z。证明了紧Hausdorff群的所有可数子群都是g-闭的，从而肯定地回答了Dikranjan，Milan和Tonolo的两个问题.证明了紧Hausdorff群的每个g-闭子群都是实紧的.本文中开发的技术被用来构造一个封闭算子g的近亲，它与紧Hausdorff阿贝尔群上的$G_δ$ -封闭相一致，从而捕获子群的实紧性和伪紧性。
Topological properties of the free topological group and the free abelian topological group on a space have been thoroughly studied since the 1940s. In this paper, we study the free topological $\mathbb{R}$-vector space $V(X)$ on $X$. We show that $V(X)$ is a quotient of the free abelian topological group on $[-1,1]\times X$, and use this to prove topological vector space analogues of existing results for free topological groups on pseudocompact spaces. As an application, we show that certain families of subspaces of $V(X)$ satisfy the so-called $\textit{algebraic colimit property}$ defined in…
Densities and Weights of Quotients of Precompact Abelian Groups
针对Banach-Mazur问题的拓扑群版本，研究准紧阿贝尔群商群的密度特征与权的最小基数及可能权值集合，给出相关基数结论、子集存在性结果，并构造反例解决了一个公开问题。
Making group topologies with, and without, convergent sequences
(1)每个无限阿贝尔紧（Hausdorff）群K允许2^ |K| -许多稠密的、非Haar-可测的基数子群|K|。当K是不可度量化的时，这些可以被选择为伪紧的。(2)每个无限阿贝尔群G都有一个2 ^2 ^ |G| -多个两两非同胚的完全有界群拓扑族A，使得G中没有非平凡序列收敛于A中的任何拓扑T. (For对于A中某些T，某些G可以安排w（G，T）< 2（|G|）。(3)每个无限阿贝尔群$G$都有一个2^2^ |G| -多个两两非同胚的完全有界群拓扑族B，其中对B中的所有T，w（G，T）= 2^ |G|，使得G中某个固定的忠实指标序列在B中的每个T中收敛到0 G.
For a Tychonoff space $X$, denote by $\mathfrak{P}$ the family of topological properties $\mathcal{P}$ of being a convergent sequence or being a compact, sequentially compact, countably compact, pseudocompact and functionally bounded subset of $X$, respectively. A maximally almost periodic $(MAP$) group $G$ respects $\mathcal{P}$ if $\mathcal{P}(G)=\mathcal{P}(G^+)$, where $G^+$ is the group $G$ endowed with the Bohr topology. We study relations between different respecting properties from $\mathfrak{P}$ and show that the respecting convergent sequences (=the Schur property) is the weakest one…
We further investigate the weak topology generated by the irreducible unitary representations of a group $G$. A deep result due to Ernest \cite{Ernest1971} and Hughes \cite{Hughes1973} asserts that every weakly compact subset of a locally compact (LC) group $G$ is compact in the LC-topology, generalizing thereby a previous result of Glicksberg \cite{glicks1962} for abelian locally compact (LCA) groups. Here, we first survey some recent findings on the weak topology and establish some new results about the preservation of several compact-like properties when going from the weak topology to the …
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