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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 families in nuclear groups

Wojciech Banaszczyk (1993)

Studia Mathematica

Nuclear groups form a class of abelian topological groups which contains LCA groups and nuclear locally convex spaces, and is closed with respect to certain natural operations. In nuclear locally convex spaces, weakly summable families are strongly summable, and strongly summable are absolutely summable. It is shown that these theorems can be generalized in a natural way to nuclear groups.

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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