scholarly journals Group reflection and precompact paratopological groups

2013 ◽  
Vol 1 ◽  
pp. 22-30 ◽  
Author(s):  
Mikhail Tkachenko
Keyword(s):  
Abelian Group ◽  
Topological Groups ◽  
Dense Subgroup ◽  
Completely Regular ◽  
Paratopological Group

AbstractWe 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, and that the group reflection, H*, of a dense subgroup G of a paratopological group G is topologically isomorphic to a subgroup of G*.

scholarly journals CONTINUITY OF MEASURABLE HOMOMORPHISMS

2008 ◽  
Vol 78 (1) ◽  
pp. 171-176 ◽  
Author(s):  
JANUSZ BRZDȨK
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Baire Property ◽  
Topological Groups ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Locally Compact Topological Group

AbstractWe give some general results concerning continuity of measurable homomorphisms of topological groups. As a consequence we show that a Christensen measurable homomorphism of a Polish abelian group into a locally compact topological group is continuous. We also obtain similar results for the universally measurable homomorphisms and the homomorphisms that have the Baire property.

scholarly journals Sequential completeness of quotient groups

2000 ◽  
Vol 61 (1) ◽  
pp. 129-150 ◽  
Author(s):  
Dikran Dikranjan ◽  
Michael Tkačenko
Keyword(s):  
Abelian Group ◽  
Direct Products ◽  
Topological Groups ◽  
Complete Group ◽  
Sequential Completeness ◽  
Algebraic Properties ◽  
Countable Compactness ◽  
Quotient Groups

We discuss various generalisations of countable compactness for topological groups that are related to completeness. The sequentially complete groups form a class closed with respect to taking direct products and closed subgroups. Surprisingly, the stronger version of sequential completeness called sequential h-completeness (all continuous homomorphic images are sequentially complete) implies pseudocompactness in the presence of good algebraic properties such as nilpotency. We also study quotients of sequentially complete groups and find several classes of sequentially q-complete groups (all quotients are sequentially complete). Finally, we show that the pseudocompact sequentially complete groups are far from being sequentially q-complete in the following sense: every pseudocompact Abelian group is a quotient of a pseudocompact Abelian sequentially complete group.

Open subgroups of free abelian topological groups

1993 ◽  
Vol 114 (3) ◽  
pp. 439-442 ◽  
Author(s):  
Sidney A. Morris ◽  
Vladimir G. Pestov
Keyword(s):  
Topological Group ◽  
Sufficient Conditions ◽  
Open Subgroup ◽  
Regular Space ◽  
Topological Groups ◽  
Covering Dimension ◽  
Abelian Topological Group ◽  
Completely Regular ◽  
Free Topological Group ◽  
Free Abelian Topological Group

We prove that any open subgroup of the free abelian topological group on a completely regular space is a free abelian topological group. Moreover, the free topological bases of both groups have the same covering dimension. The prehistory of this result is as follows. The celebrated Nielsen–Schreier theorem states that every subgroup of a free group is free, and it is equally well known that every subgroup of a free abelian group is free abelian. The analogous result is not true for free (abelian) topological groups [1,5]. However, there exist certain sufficient conditions for a subgroup of a free topological group to be topologically free [2]; in particular, an open subgroup of a free topological group on a kω-space is topologically free. The corresponding question for free abelian topological groups asked 8 years ago by Morris [11] proved to be more difficult and remained open even within the realm of kω-spaces. In the present paper a comprehensive answer to this question is obtained.

scholarly journals Invariant metrics on free topological groups

1973 ◽  
Vol 9 (1) ◽  
pp. 83-88 ◽  
Author(s):  
Sidney A. Morris ◽  
H.B. Thompson
Keyword(s):  
Topological Group ◽  
Invariant Metrics ◽  
Regular Space ◽  
Discrete Topology ◽  
Group Topology ◽  
Topological Groups ◽  
Abstract Group ◽  
Completely Regular ◽  
Free Topological Group ◽  
Totally Disconnected

For a completely regular space X, G(X) denotes the free topological group on X in the sense of Graev. Graev proves the existence of G(X) by showing that every pseudo-metric on X can be extended to a two-sided invariant pseudo-metric on the abstract group G(X). It is natural to ask if the topology given by these two-sided invariant pseudo-metrics on G(X) is precisely the free topological group topology on G(X). If X has the discrete topology the answer is clearly in the affirmative. It is shown here that if X is not totally disconnected then the answer is always in the negative.

scholarly journals When are dense subgroups of LCA groups closed under intersections?

1994 ◽  
Vol 49 (1) ◽  
pp. 59-67
Author(s):  
M.A. Khan
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Compact Abelian Group ◽  
Additive Group ◽  
Locally Compact Abelian Group ◽  
Torsion Subgroup ◽  
Dense Subgroup ◽  
Locally Compact ◽  
Dense Subgroups ◽  
Lca Groups

Let G be a nondiscrete locally compact Hausdorff abelian group. It is shown that if G contains an open torsion subgroup, then every proper dense subgroup of G is contained in a maximal subgroup; while if G has no open torsion subgroup, then it has a dense subgroup D such that G/D is algebraically isomorphic to R, the additive group of reals. With each G, containing an open torsion subgroup, we associate the least positive integer n such that the nth multiple of every discontinuous character of G is continuous. The following are proved equivalent for a nondiscrete locally compact abelian group G:(1) The intersection of any two dense subgroups of G is dense in G.(2) The intersection of all dense subgroups of G is dense in G.(3) G contains an open torsion subgroup, and for each prime p dividing the positive integer associated with G, pG is either open or a proper dense subgroup of G.Finally, we construct a locally compact abelian group G with infinitely many dense subgroups satisfying the three equivalent conditions stated above.

scholarly journals A Theorem on Algebras of Measures on Topological Groups

1959 ◽  
Vol 11 (4) ◽  
pp. 195-206 ◽  
Author(s):  
J. H. Williamson
Keyword(s):  
Banach Space ◽  
Abelian Group ◽  
Compact Abelian Group ◽  
Locally Compact Abelian Group ◽  
Topological Groups ◽  
Finite Sets ◽  
Locally Compact ◽  
Borel Measures ◽  
Bounded Complex ◽  
Image Position

Let G be a locally compact Abelian group, and the set of bounded complex (regular countably-additive Borel) measures on G. It is well known that becomes a Banach space if the norm is defined bythe supremum being over all finite sets of disjoint Borel subsets of G.

scholarly journals Cardinal invariants of paratopological groups

2013 ◽  
Vol 1 ◽  
pp. 37-45 ◽  
Author(s):  
Iván Sánchez
Keyword(s):  
Countable Family ◽  
Completely Regular ◽  
Cardinal Invariants ◽  
Paratopological Group ◽  
Weak Lindelöf Number ◽  
Lindelöf Number

AbstractWe show that a regular totally ω-narrow paratopological group G has countable index of regularity, i.e., for every neighborhood U of the identity e of G, we can find a neighborhood V of e and a countable family of neighborhoods of e in G such that ∩W∈γ VW−1⊆ U. We prove that every regular (Hausdorff) totally !-narrow paratopological group is completely regular (functionally Hausdorff). We show that the index of regularity of a regular paratopological group is less than or equal to the weak Lindelöf number. We also prove that every Hausdorff paratopological group with countable π- character has a regular Gσ-diagonal.

Varieties of topological groups III

1970 ◽  
Vol 2 (2) ◽  
pp. 165-178 ◽  
Author(s):  
Sidney A. Morris
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Algebraic Variety ◽  
Additive Group ◽  
Topological Groups ◽  
Locally Compact ◽  
Circle Group ◽  
Free Topological Group ◽  
The Family ◽  
Usual Topology

This paper continues the invèstigation of varieties of topological groups. It is shown that the family of all varieties of topological groups with any given underlying algebraic variety is a class and not a set. In fact the family of all β-varieties with any given underlying algebraic variety is a class and not a set. A variety generated by a family of topological groups of bounded cardinal is not a full variety.The varieties V(R) and V(T) generated by the additive group of reals and the circle group respectively each with its usual topology are examined. In particular it is shown that a locally compact Hausdorff abelian group is in V(T) if and only if it is compact. Thus V(R) properly contains V(T).It is proved that any free topological group of a non-indiscrete variety is disconnected. Finally, some comments are made on topologies on free groups.

scholarly journals Factoring Continuous Characters Defined on Subgroups of Products of Topological Groups †

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 167
Author(s):  
Mikhail G. Tkachenko
Keyword(s):  
Typical Result ◽  
Topological Groups ◽  
Dense Subgroup ◽  
Finite Set ◽  
Continuous Character

This study is on the factorization properties of continuous homomorphisms defined on subgroups (or submonoids) of products of (para)topological groups (or monoids). A typical result is the following one: Let D=∏i∈IDi be a product of paratopological groups, S be a dense subgroup of D, and χ a continuous character of S. Then one can find a finite set E⊂I and continuous characters χi of Di, for i∈E, such that χ=∏i∈Eχi∘piS, where pi:D→Di is the projection.

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