Fine convergence in free groups
Foliations induced by congruences.
Free non-archimedean topological groups
We study free topological groups defined over uniform spaces in some subclasses of the class of non-archimedean groups. Our descriptions of the corresponding topologies show that for metrizable uniformities the corresponding free balanced, free abelian and free Boolean groups are also metrizable. Graev type ultra-metrics determine the corresponding free topologies. Such results are in a striking contrast with free balanced and free abelian topological groups cases (in standard varieties). Another...
Free products of topological groups with equal uniformities, I
Free products of topological groups with equal uniformities, II
Fuzzy neighborhood structures on partially ordered groups.
Generalization of a theorem of Bogolioùboff to topological groups. With an appendix on Banach mean values in nonabelian groups
Group reflection and precompact paratopological groups
We construct a precompact completely regular paratopological Abelian group G of size (2ω)+ such that all subsets of G of cardinality ≤ 2ω are closed. This shows that Protasov’s theorem on non-closed discrete subsets of precompact topological groups cannot be extended to paratopological groups. We also prove that the group reflection of the product of an arbitrary family of paratopological (even semitopological) groups is topologically isomorphic to the product of the group reflections of the factors,...
Group structure in finite coverings of compact solenoidal groups.
Group Structures and Rectifiability in Powers of Spaces
We prove that if some power of a space X is rectifiable, then is rectifiable. It follows that no power of the Sorgenfrey line is a topological group and this answers a question of Arhangel’skiĭ. We also show that in Mal’tsev spaces of point-countable type, character and π-character coincide.
Groups – Additive Notation
We translate the articles covering group theory already available in the Mizar Mathematical Library from multiplicative into additive notation. We adapt the works of Wojciech A. Trybulec [41, 42, 43] and Artur Korniłowicz [25]. In particular, these authors have defined the notions of group, abelian group, power of an element of a group, order of a group and order of an element, subgroup, coset of a subgroup, index of a subgroup, conjugation, normal subgroup, topological group, dense subset and basis...
Gruppentopologien auf nichtabelschen abzählbaren Gruppen.
Homogeneity of βG if G is a topological group
Homomorphic images of -factorizable groups
It is well known that every -factorizable group is -narrow, but not vice versa. One of the main problems regarding -factorizable groups is whether this class of groups is closed under taking continuous homomorphic images or, alternatively, whether every -narrow group is a continuous homomorphic image of an -factorizable group. Here we show that the second hypothesis is definitely false. This result follows from the theorem stating that if a continuous homomorphic image of an -factorizable...
Idempotents and the multiplicative group of some totally bounded rings
In this paper, we extend some results of D. Dolzan on finite rings to profinite rings, a complete classification of profinite commutative rings with a monothetic group of units is given. We also prove the metrizability of commutative profinite rings with monothetic group of units and without nonzero Boolean ideals. Using a property of Mersenne numbers, we construct a family of power commutative non-isomorphic profinite semiprimitive rings with monothetic group of units.
Identifying and distinguishing various varieties of abelian topological groups [Book]
In quest of weaker connected topologies
We study when a topological space has a weaker connected topology. Various sufficient and necessary conditions are given for a space to have a weaker Hausdorff or regular connected topology. It is proved that the property of a space of having a weaker Tychonoff topology is preserved by any of the free topological group functors. Examples are given for non-preservation of this property by “nice” continuous mappings. The requirement that a space have a weaker Tychonoff connected topology is rather...
Inversion and joint continuity in semigroups on kw-spaces.
Irresolvable countable spaces of weight less than
We construct in Bell-Kunen’s model: (a) a group maximal topology on a countable infinite Boolean group of weight and (b) a countable irresolvable dense subspace of . In this model .
Large families of dense pseudocompact subgroups of compact groups
We prove that every nonmetrizable compact connected Abelian group G has a family H of size |G|, the maximal size possible, consisting of proper dense pseudocompact subgroups of G such that H ∩ H'={0} for distinct H,H' ∈ H. An easy example shows that connectedness of G is essential in the above result. In the general case we establish that every nonmetrizable compact Abelian group G has a family H of size |G| consisting of proper dense pseudocompact subgroups of G such that each intersection H H'...