On topological groups with a first-countable remainder,III

We prove a general theorem that allows us to conclude that under CH, the free topological group over a nontrivial convergent sequence S$S$ has a first-countable remainder. It is also shown that any separable non-metrizable topological group with a first-countable remainder is Rajkov complete.     

Roughness and approximation of quasi-representations of amenable groups

Sufficient conditions are obrained for a quasi-representation (not necessarily bounded) of an amenable group (topological in general) to be a bounded perturbation of an ordinary representation. In particular, it is shown that an arbitrary (not necessarily bounded) finite-dimensional quasi-representation of an amenable topological group is a bounded perturbation of an ordinary representation. Translated fromMatematicheskie Zametki, Vol. 65, No. 6, pp. 908–920, June, 1999.     

Continuous Selections and Locally Pseudocompact Groups

We characterize locally pseudocompact groups by means of the selection theory. Our result is the selection version of the well-known Comfort—Ross theorem on pseudocompactness which states that a topological group is pseudocompact if and only its Stone—Čech compactification is a topological group.     

Strong Reflexivity of Abelian Groups

A reflexive topological group G is called strongly reflexive if each closed subgroup and each Hausdorff quotient of the group G and of its dual group is reflexive. In this paper we establish an adequate concept of strong reflexivity for convergence groups. We prove that complete metrizable nuclear groups and products of countably many locally compact topological groups are BB-strongly reflexive.     

Strictly localizable measures

It is proved that, under the Continuum Hypotesis (CH), every complete, locally determined, Radon measures of type (H), on a topological space with countable basis, is strictly localizable. This result is useful in the theory of invariant measures on a topological group and, in particular, in the theory of Hausdorff measures.     

关于几乎拓扑群的注记

In this paper, we mainly discuss some generalized metric properties and the cardinal invariants of almost topological groups. We give a characterization for an almost topological group to be a topological group and show that:(1) Each almost topological group that is of countable π-character is submetrizable;(2) Each left λ-narrow almost topological group isλ-narrow;(3) Each separable almost topological group is ω-narrow. Some questions are posed.     

代数结构上的模糊拟伪$b$-度量

研究代数结构上的模糊(拟)伪$b$-度量. 主要结论有: (1)设$G$是一个抽象群, $\tau$是$G$上一个左不变模糊拟伪$b$-度量(伪$b$-度量)诱导的拓扑,如果$(G,\tau)$是右拓扑群,那么$(G,\tau)$是一个仿拓扑群(拓扑群); (2)设$S$是一个半群,如果$\tau$是$S$上一个不变模糊拟伪$b$-度量诱导的拓扑,那么$(S,\tau)$是一个拓扑半群.     

Transversals in non-discrete groups

The concept of ‘topological right transversal’ is introduced to study right transversals in topological groups. Given any right quasigroupS with a Tychonoff topologyT, it is proved that there exists a Hausdorff topological group in whichS can be embedded algebraically and topologically as a right transversal of a subgroup (not necessarily closed). It is also proved that if a topological right transversal(S, T _S ,T ^S , o) is such thatT _S =T ^S is a locally compact Hausdorff topology onS, thenS can be embedded as a right transversal of a closed subgroup in a Hausdorff topological group which is universal in some sense.     

The <Emphasis Type="Italic">C</Emphasis>-topology on lattice-ordered groups

Let A be a lattice-ordered group. Gusi′c showed that A can be equipped with a C-topology which makes A into a topological group. We give a generalization of Gusi′c's theorem,and reveal the very nature of a "C-group" of Gusi′c in this paper. Moreover,we show that the C-topological groups are topological lattice-ordered groups,and prove that every archimedean lattice-ordered vector space is a T2 topological lattice-ordered vector space under the C-topology. An easy example shows that a C-group need not be T2....     

Topological classification of supertransitive flows on closed nonorientable surfaces. Part I. Necessary conditions for topological equivalence

In this paper a topological invariant is introduced for the class of supertransitive flows on closed nonorientable surfacesM of negative Euler characteristic. We describe properties of this invariant and prove that it provides necessary conditions for the topological equivalence of flows belonging to the above-mentioned class of supertransitive flows. Translated fromMatematicheskie Zametki, Vol. 68, No. 3, pp. 461–470, September, 2000.     

The structure of abelian pro-Lie groups

A pro-Lie group is a projective limit of a projective system of finite dimensional Lie groups. A prodiscrete group is a complete abelian topological group in which the open normal subgroups form a basis of the filter of identity neighborhoods. It is shown here that an abelian pro-Lie group is a product of (in general infinitely many) copies of the additive topological group of reals and of an abelian pro-Lie group of a special type; this last factor has a compact connected component, and a characteristic closed subgroup which is a union of all compact subgroups; the factor group modulo this subgroup is pro-discrete and free of nonsingleton compact subgroups. Accordingly, a connected abelian pro-Lie group is a product of a family of copies of the reals and a compact connected abelian group. A topological group is called compactly generated if it is algebraically generated by a compact subset, and a group is called almost connected if the factor group modulo its identity component is compact. It is further shown that a compactly generated abelian pro-Lie group has a characteristic almost connected locally compact subgroup which is a product of a finite number of copies of the reals and a compact abelian group such that the factor group modulo this characteristic subgroup is a compactly generated prodiscrete group without nontrivial compact subgroups.Mathematics Subject Classification (1991): 22B, 22E     

混沌群作用

§ 1　ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｉｓｐａｐｅｒｉｓｔｈｅｓｅｑｕｅｌｔｏ[1 ,2 ].Ｃａｉｒｎｓｅｔａｌ.[1] ｉｎｔｒｏｄｕｃｅｄｔｈｅｎｏｔｉｏｎｏｆａｃｈａｏｔｉｃｇｒｏｕｐａｃｔｉｏｎａｓａｇｅｎｅｒａｌｉｚａｔｉｏｎｏｆｃｈａｏｔｉｃｄｙｎａｍｉｃａｌｓｙｓｔｅｍｓ(ｓｅｅｄｅｆｉｎｉｔｉｏｎｂｅｌｏｗ) .Ｔｈｅｙｓｈｏｗｅｄｔｈａｔｔｈｅｃｉｒｃｌｅｄｏｅｓｎｏｔａｄｍｉｔａｃｈａｏｔｉｃａｃｔｉｏｎｏｆａｎｙｇｒｏｕｐ ,ａｎｄｃｏｎｓｔｒｕｃｔｅｄａｃｈａｏｔｉｃａｃｔｉｏｎｏｆＧ =Ｚ×…     

Topological Hypergroups in the Sense of Marty

In this paper, we introduce the concept of topological hypergroups as a generalization of topological groups. A topological hypergroup is a nonempty set endowed with two structures, that of a topological space and that of a hypergroup. Let (H, ○) be a hypergroup and (H, τ) be a topological space such that the mappings (x, y) → x ○ y and (x, y) → x/y from H × H to 𝒫*(H) are continuous. The main tool to obtain basic properties of hypergroups is the fundamental relation β*. So, by considering the quotient topology induced by the fundamental relation on a hypergroup (H, ○) we show that if every open subset of H is a complete part, then the fundamental group of H is a topological group.

It is important to mention that in this paper the topological hypergroups are different from topological hypergroups which was initiated by Dunkl and Jewett.     

拓扑群在连续统上的膨胀作用

本文把一个同胚的膨胀作用推广到拓扑群的情形,并研究了有限生成离散群 的膨胀作用,得到了如下结果:Z×Z不能膨胀地作用在单位闭区间I上,而自由积 Z★Z可以膨胀地作用在I上.     

On the smoothness of some quotients of Banach-Lie groups

Quotients of Banach-Lie groups are regarded as topological groups with Lie algebra in the sense of Hofmann-Morris on the one hand, and as Q-groups in the sense of Barre-Plaisant on the other hand. For the groups of the type $G/N$ where $N\subseteq G$ is a pseudo-discrete normal subgroup, their Lie algebra in the sense of Q-groups turns out to be isomorphic to the Lie algebra of G, which is in general merely a dense subalgebra of the Lie algebra of $G/N$ when regarded as a topological group with Lie algebra. The submersion-like behavior of quotient maps of Banach-Lie groups is also investigated. The two aforementioned approaches to the Lie theory of the quotients of Banach-Lie groups thus lead to differing results and the Lie theoretic properties of quotient groups are more accurately described by the Q-group approach than by the approach via topological groups with Lie algebras.     

Three-Dimensional Topological Loops with Solvable Multiplication Groups

We prove that each 3-dimensional connected topological loop L having a solvable Lie group of dimension ≤5 as the multiplication group of L is centrally nilpotent of class 2. Moreover, we classify the solvable non-nilpotent Lie groups G which are multiplication groups for 3-dimensional simply connected topological loops L and dim G ≤ 5. These groups are direct products of proper connected Lie groups and have dimension 5. We find also the inner mapping groups of L.     

Topological games and continuity of group operations

We consider a topological game GΠ involving two players α and β and show that, for a paratopological group, the absence of a winning strategy for player β implies the group is a topological one. We provide a large class of topological spaces X for which the absence of a winning strategy for player β is equivalent to the requirement that X is a Baire space. This allows to extend the class of paratopological or semitopological groups for which one can prove that they are, actually, topological groups.Conditions of the type “existence of a winning strategy for the player α” or “absence of a winning strategy for the player β” are frequently used in mathematics. Though convenient and satisfactory for theoretical considerations, such conditions do not reveal much about the internal structure of the topological space where they hold. We show that the existence of a winning strategy for any of the players in all games of Banach-Mazur type can be expressed in terms of “saturated sieves” of open sets.     

HFD groups in the Solovay model

Hajnal and Juhász proved that under CH there is a hereditarily separable, hereditarily normal topological group without non-trivial convergent sequences that is countably compact and not Lindelöf. The example constructed is a topological subgroup H⊆ω₁² that is an HFD with the following property

(P)

the projection of H onto every partial product I² for I∈ω[ω₁] is onto.

Any such group has the necessary properties. We prove that if κ is a cardinal of uncountable cofinality, then in the model obtained by forcing over a model of CH with the measure algebra on κ², there is an HFD topological group in ω₁² which has property (P).     

On the Problem of Condensation onto Compact Spaces

Doklady Mathematics - Assuming the continuum hypothesis CH, it is proved that there exists a perfectly normal compact topological space Z and a countable set $$E \subset Z$$ such that...