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Semi-quotient mappings and spaces

Moiz ud Din Khan, Rafaqat Noreen, Muhammad Siddique Bosan (2016)

Open Mathematics

In this paper, we continue the study of s-topological and irresolute-topological groups. We define semi-quotient mappings which are stronger than semi-continuous mappings, and then consider semi-quotient spaces and groups. It is proved that for some classes of irresolute-topological groups (G, *, τ) the semi-quotient space G/H is regular. Semi-isomorphisms of s-topological groups are also discussed.

Some aspects of nuclear vector groups

Lydia Außenhofer (2001)

Studia Mathematica

In [2] W. Banaszczyk introduced nuclear groups, a Hausdorff variety of abelian topological groups which is generated by all nuclear vector groups (cf. 2.3) and which contains all nuclear vector spaces and all locally compact abelian groups. We prove in 5.6 that the Hausdorff variety generated by all nuclear vector spaces and all locally compact abelian groups (denoted by 𝒱₁) is strictly smaller than the Hausdorff variety of all nuclear groups (denoted by 𝒱₂). More precisely,...

Stabilizers of closed sets in the Urysohn space

Julien Melleray (2006)

Fundamenta Mathematicae

Building on earlier work of Katětov, Uspenskij proved in [8] that the group of isometries of Urysohn's universal metric space 𝕌, endowed with the pointwise convergence topology, is a universal Polish group (i.e. it contains an isomorphic copy of any Polish group). Answering a question of Gao and Kechris, we prove here the following, more precise result: for any Polish group G, there exists a closed subset F of 𝕌 such that G is topologically isomorphic to the group of isometries of 𝕌 which map...

Strong reflexivity of Abelian groups

Montserrat Bruguera, María Jesús Chasco (2001)

Czechoslovak Mathematical Journal

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.

Subgroups and products of ℝ -factorizable P -groups

Constancio Hernández, Mihail G. Tkachenko (2004)

Commentationes Mathematicae Universitatis Carolinae

We show that every subgroup of an ℝ -factorizable abelian P -group is topologically isomorphic to a closed subgroup of another ℝ -factorizable abelian P -group. This implies that closed subgroups of ℝ -factorizable P -groups are not necessarily ℝ -factorizable. We also prove that if a Hausdorff space Y of countable pseudocharacter is a continuous image of a product X = ∏ i ∈ I X i of P -spaces and the space X is pseudo- ω 1 -compact, then n w ( Y ) ≤ ℵ 0 . In particular, direct products of ℝ -factorizable P -groups are ℝ -factorizable and...

Subgroups of ℝ -factorizable groups

Constancio Hernández, Mihail G. Tkachenko (1998)

Commentationes Mathematicae Universitatis Carolinae

The properties of ℝ -factorizable groups and their subgroups are studied. We show that a locally compact group G is ℝ -factorizable if and only if G is σ -compact. It is proved that a subgroup H of an ℝ -factorizable group G is ℝ -factorizable if and only if H is z -embedded in G . Therefore, a subgroup of an ℝ -factorizable group need not be ℝ -factorizable, and we present a method for constructing non- ℝ -factorizable dense subgroups of a special class of ℝ -factorizable groups. Finally, we construct a closed...

Summable Family in a Commutative Group

Roland Coghetto (2015)

Formalized Mathematics

Hölzl et al. showed that it was possible to build “a generic theory of limits based on filters” in Isabelle/HOL [22], [7]. In this paper we present our formalization of this theory in Mizar [6]. First, we compare the notions of the limit of a family indexed by a directed set, or a sequence, in a metric space [30], a real normed linear space [29] and a linear topological space [14] with the concept of the limit of an image filter [16]. Then, following Bourbaki [9], [10] (TG.III, §5.1 Familles sommables...

Sur le groupe des difféomorphismes du tore

Michael R. Herman (1973)

Annales de l'institut Fourier

Il est démontré que le groupe des difféomorphismes C ∞ du tore qui sont C ∞ isotopes à l’identité est un groupe qui est égal à son groupe des commutateurs. Il résulte de D.A.B. Epstein que c’est un groupe simple. Un lemme fondamental est utilisé ; il donne la structure locale des orbites de certaines translations du tore ; ce lemme est une application du théorème des fonctions implicites de F. Sergeraert.

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