局部伪紧群的零维性与连通分支

函数可数子代数与Banaschewski紧化的若干性质

Functionally countable subalgebras and some properties of Banaschewski compactification

设$X$是一个零维空间，$C_c(X)$是$X$上所有具有可数象的连续实值函数的集合。在本文中，我们用$C_c^K(X)$表示（分别，$C_{c}^ψ(X)$）$C_c(X)$中所有函数的集合，伪紧）支持。首先，我们观察到$C_c^{K}(X)=O_c^{β_0X\setminus X}$（分别为，$C_c^ψ(X)=M_c^{β_0X\setminus \upsilon_0X}$）。这意味着对于$\Bbb{N}$ -紧空间$X$，$C_c(X)$中所有自由极大理想的交集等于$C_c^K(X)$，即，$M_c^{β_0X\setminus X}=C_c^K(X)$。然后，利用函数可数子代数的方法，得到了Banaschewski紧化余项的一些结果。证明了对零维非伪紧空间$X$，集合$β_0X\setminus \upsilon_0X$的基数至少为$2^{2^{\aleph_0}}$ .此外，对于局部紧的$\Bbb{N}$ -紧空间$X$，余项$β_0X\setminus X$是几乎$P$ -空间.这些结果使我们在非伪紧零维空间的Banaschewski紧化中找到了一类Parovi $\breve{\mbox{c}}$ enko空间.我们的结论是一个定理，给出了一个下界的细胞的子空间$β_0X\setminus \upsilon_0X$和$β_0X\setminus X$，只要$X$是…

Quotient with respect to admissible $L$-subgyrogroups

The concept of gyrogroups, with a weaker algebraic structure without associative law, was introduced under the background of $c$-ball of relativistically admissible velocities with Einstein velocity addition. A topological gyrogroup is just a gyrogroup endowed with a compatible topology such that the multiplication is jointly continuous and the inverse is continuous. This concept is a good generalization of a topological group. In this paper, we are going to establish that for a locally compact admissible $L$-subgyrogroup $H$ of a strongly topological gyrogroup $G$, the natural quotient mappin…

The weak compactification of locally compact groups

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 …

逆的连续性

Continuity of the inverse

该研究定义了Δ-Baire空间，证明满足该条件的仿拓扑群是拓扑群，还指出多类经典拓扑空间属于Δ-Baire空间，推进了拓扑群相关理论。

拓扑群的子群的稠密性character

Density character of subgroups of topological groups

可分度量空间X的子空间Y是可分的，但如果X不是可度量的，即使Y是拓扑向量空间X的闭线性子空间，这也不是真的。K.H. Hofmann和S.A. Morris引入了由有限维李群的投射极限组成的亲李群类，证明了它包含所有紧群、局部紧阿贝尔群和连通局部紧群，并且在乘积和闭子群下是闭的.一个拓扑群G是几乎连通的，如果G由其单位元的连通分支所构成的商群是紧的。证明了几乎连通的原李群是可分的当且仅当它的权不大于c。证明了几乎连通原李群是可分的当且仅当它是可分T2空间的子空间。证明了局部紧（偶羽）拓扑群G是可分Hausdorff拓扑群的子群，但如果仅假设G同胚于可分Tychonoff空间的子空间，则结论是错误的.每个权小于或等于c的预紧拓扑群拓扑同构于权为c的可分伪紧群的闭子群。这意味着可分伪紧群有大量的闭不可分子群。在CH下给出了一个包含不可分闭子群的可分可数紧阿贝尔群的例子。证明了Ω-窄拓扑群G的下列条件是等价的：（i）G是可分正则空间的子空间;（ii）G是可分拓扑群的子群;（iii）G是可分路径连通局部路径连通群的闭子群.

The Separable Quotient Problem for Topological Groups

The famous Banach-Mazur problem, which asks if every infinite-dimensional Banach space has an infinite-dimensional separable quotient Banach space, has remained unsolved for 85 years, though it has been answered in the affirmative for reflexive Banach spaces and even Banach spaces which are duals. The analogous problem for locally convex spaces has been answered in the negative, but has been shown to be true for large classes of locally convex spaces including all non-normable Fréchet spaces. In this paper the analogous problem for topological groups is investigated. Indeed there are four natu…

局部预紧群：（局部）实紧性与连通性

Locally precompact groups: (Local) realcompactness and connectedness

A. Weil断言一个拓扑群嵌入一个局部紧群的（稠密）子群当且仅当它包含一个非空的预紧开集;这样的群被称为局部预紧群。在局部预紧群类中，作者将具有以下拓扑性质的群分类：迪厄多内完备性;局部实紧性;实紧性;遗传实紧性;连通性;局部连通性;零维性。他们还证明了一个阿贝尔局部预紧群作为一个拓扑群的拟分支出现当且仅当它是预紧生成的，也就是说，它是由一个预紧子集代数生成的。

Products of topological groups in which all closed subgroups are separable

We prove that if $H$ is a topological group such that all closed subgroups of $H$ are separable, then the product $G\times H$ has the same property for every separable compact group $G$. Let $c$ be the cardinality of the continuum. Assuming $2^{ω_1} = c$, we show that there exist: (1) pseudocompact topological abelian groups $G$ and $H$ such that all closed subgroups of $G$ and $H$ are separable, but the product $G\times H$ contains a closed non-separable $σ$-compact subgroup; (2) pseudocomplete locally convex vector spaces $K$ and $L$ such that all closed vector subspaces of $K$ and $L$ are s…

$C_p(X)$中代数结构的拓扑视角

A topological view of algebraic structures in $C_p(X)$

该研究从拓扑角度刻画$C_p(X)$中理想闭包，关联$C(X)$子环稠密性与分点性，还明确$C_p(X)$相关性质对应Tychonoff空间$X$的拓扑特征。

General Non-Commutative Locally Compact Locally Hausdorff Stone Duality

We extend the classical Stone duality between zero dimensional compact Hausdorff spaces and Boolean algebras. Specifically, we simultaneously remove the zero dimensionality restriction and extend to étale groupoids, obtaining a duality with an elementary class of inverse semigroups.

SPM公告27

SPM Bulletin 27

内容：1.编者按2.研究报告2.1没有无限紧子集的伪紧群拓扑2.2.（FIN[1]k）上的选择余理想2.3.整函数映射不可数稠密集的实数到对方单调2.4。Borel覆盖层的有效精炼2.5.对称和着色：一些结果和开放问题2.6.密度2.7以下的许多配分关系。林德洛夫不灭性、拓扑博弈与选择原理2.8。局部预紧群：（局部）实紧性和连通性2.9。群Aut是Roelcke预紧2.10。带偏序集参数的无限组合语句2.11.滤波器的舒尔1定理2.12.局部紧阿贝尔群中的一致渐近上密度2.13.一个c 0-饱和巴拿赫空间没有长的无条件基本序列2.14. 1上多项式的零子空间（2.15. MAD Families和SANE Player 2.16.关于d_k > cov_k（M）的一致性2.17.自由拓扑群的o-有界性2.18.与超滤子的正则性和拓扑空间的紧性有关的组合和模型理论原理。V 2.19. Sorgenfrey线的Menger子集2.20.与超滤子的正则性和拓扑空间的紧性有关的组合和模型理论原理。六.早期问题未解决的问题

准紧阿贝尔群的商群的密度与权

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。