A reflexive topological group is called strongly reflexive if each closed subgroup and each Hausdorff quotient of the group 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.
We show that every subgroup of an -factorizable abelian -group is topologically isomorphic to a closed subgroup of another -factorizable abelian -group. This implies that closed subgroups of -factorizable -groups are not necessarily -factorizable. We also prove that if a Hausdorff space of countable pseudocharacter is a continuous image of a product of -spaces and the space is pseudo--compact, then . In particular, direct products of -factorizable -groups are -factorizable and...
The properties of -factorizable groups and their subgroups are studied. We show that a locally compact group is -factorizable if and only if is -compact. It is proved that a subgroup of an -factorizable group is -factorizable if and only if is -embedded in . Therefore, a subgroup of an -factorizable group need not be -factorizable, and we present a method for constructing non--factorizable dense subgroups of a special class of -factorizable groups. Finally, we construct a closed...
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.
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...
Il est démontré que le groupe des difféomorphismes du tore qui sont 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.