A locally quasi-convex abelian group without a Mackey group topology

2018 ◽  
Vol 146 (8) ◽  
pp. 3627-3632 ◽  
Author(s):  
Saak Gabriyelyan
Keyword(s):  
Abelian Group ◽  
Group Topology

scholarly journals A note on the duality of locally compact groups

1968 ◽  
Vol 9 (2) ◽  
pp. 87-91 ◽  
Author(s):  
J. W. Baker
Keyword(s):  
Abelian Group ◽  
Compact Group ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Group Topology ◽  
Locally Compact ◽  
Natural Way

Let H be a group of characters on an (algebraic) abelian group G. In a natural way, we may regard G as a group of characters on H. In this way, we obtain a duality between the two groups G and H. One may pose several problems about this duality. Firstly, one may ask whether there exists a group topology on G for which H is precisely the set of continuous characters. This question has been answered in the affirmative in [1]. We shall say that such a topology is compatible with the duality between G and H. Next, one may ask whether there exists a locally compact group topology on G which is compatible with a given duality and, if so, whether there is more than one such topology. It is this second question (previously considered by other authors, to whom we shall refer below) which we shall consider here.

The Meet Operator in the Lattice of Group Topologies

1986 ◽  
Vol 29 (4) ◽  
pp. 478-481
Author(s):  
Bradd Clark ◽  
Victor Schneider
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Locally Compact Group ◽  
Group Topology ◽  
Locally Compact ◽  
Hausdorff Topology ◽  
Lattice Of Topologies ◽  
Group Form ◽  
Complemented Lattice ◽  
Infinite Abelian Group

AbstractIt is well known that the lattice of topologies on a set forms a complete complemented lattice. The set of topologies which make G into a topological group form a complete lattice L(G) which is not a sublattice of the lattice of all topologies on G.Let G be an infinite abelian group. No nontrivial Hausdorff topology in L(G) has a complement in L(G). If τ1 and τ2 are locally compact topologies then τ1Λτ2 is also a locally compact group topology. The situation when G is nonabelian is also considered.

scholarly journals TΩ-sequences in abelian groups

2000 ◽  
Vol 24 (3) ◽  
pp. 145-148 ◽  
Author(s):  
Robert Ledet ◽  
Bradd Clark
Keyword(s):  
Abelian Group ◽  
Fundamental System ◽  
Abelian Groups ◽  
Group Topology ◽  
Hausdorff Group

A sequence in an abelian group is called aT-sequence if there exists a Hausdorff group topology in which the sequence converges to zero. This paper describes the fundamental system for the finest group topology in which this sequence converges to zero. A sequence is aTΩ-sequence if there exist uncountably many different Hausdorff group topologies in which the sequence converges to zero. The paper develops a condition which insures that a sequence is aTΩ-sequence and examples ofTΩ-sequences are given.

scholarly journals Algebraic obstructions to sequential convergence in Hausdorrf abelian groups

1998 ◽  
Vol 21 (1) ◽  
pp. 93-96 ◽  
Author(s):  
Bradd Clark ◽  
Sharon Cates
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Group Topology ◽  
Sequential Convergence ◽  
Hausdorff Group

Given an abelian groupGand a non-trivial sequence inG, when will it be possible to construct a Hausdroff topology onGthat allows the sequence to converge? As one might expect of such a naive question, the answer is far too complicated for a simple response. The purpose of this paper is to provide some insights to this question, especially for the integers, the rationals, and any abelian groups containing these groups as subgroups. We show that the sequence of squares in the integers cannot converge to0in any Hausdroff group topology. We demonstrate that any sequence in the rationals that satisfies a “sparseness” condition will converge to0in uncountably many different Hausdorff group topologies.

Locally compact topologies on abelian groups

1987 ◽  
Vol 101 (2) ◽  
pp. 233-235 ◽  
Author(s):  
Sidney A. Morris
Keyword(s):  
Abelian Group ◽  
Compact Group ◽  
Infinite Product ◽  
Locally Compact Group ◽  
Group Topology ◽  
Topological Groups ◽  
Real Numbers ◽  
Locally Compact ◽  
Cyclic Groups ◽  
Totally Disconnected

AbstractIt is shown that an abelian group admits a non-discrete locally compact group topology if and only if it has a subgroup algebraically isomorphic to the group of p-adic integers or to an infinite product of non-trivial finite cyclic groups. It is also proved that an abelian group admits a non-totally-disconnected locally compact group topology if and only if it has a subgroup algebraically isomorphic to the group of real numbers. Further, if an abelian group admits one non-totally-disconnected locally compact group topology then it admits a continuum of such topologies, no two of which yield topologically isomorphic topological groups.

scholarly journals Continuity properties of discontinuous homomorphisms and refinements of group topologies

Filomat ◽  
2017 ◽  
Vol 31 (7) ◽  
pp. 2183-2188
Author(s):  
H.J. Bello ◽  
L. Rodríguez ◽  
M.G. Tkachenko
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Group Topology ◽  
Topological Groups ◽  
Continuity Properties ◽  
Accumulation Points

We present several conditions on topological groups G and H under which every discontinuous homomorphism of G to H preserves accumulation points of open sets in G. It is also proved that every (locally) precompact abelian group admits a strictly finer zero-dimensional (locally) precompact topological group topology of the same weight as the original one.

scholarly journals Every uncountable abelian group admits a nonnormal group topology

1994 ◽  
Vol 122 (3) ◽  
pp. 907-907 ◽  
Author(s):  
F. Javier Trigos-Arrieta
Keyword(s):  
Abelian Group ◽  
Group Topology

A theorem on the Bohr compactification of a locally compact Abelian group

1965 ◽  
Vol 61 (1) ◽  
pp. 65-68 ◽  
Author(s):  
N. Th. Varopoulos
Keyword(s):  
Abelian Group ◽  
Cardinal Number ◽  
Locally Compact Group ◽  
Locally Compact Abelian Group ◽  
Dimensional Vector ◽  
Group Topology ◽  
Vector Group ◽  
Locally Compact ◽  
Bohr Compactification ◽  
Character Group

Notations. If G denotes an Abelian group and L a locally compact group topology on G, G(L) will denote the resulting topological group, and will denote the character group with the usual Pontrjagin topology (which is locally compact). Rn will denote the real n-dimensional vector group. Finally, for any set X, |X| will denote its cardinal number.

A group topology on the free abelian group of cardinality c that makes its square countably compact

2011 ◽  
Vol 212 (3) ◽  
pp. 235-260 ◽  
Author(s):  
Ana Carolina Boero ◽  
Artur Hideyuki Tomita
Keyword(s):  
Abelian Group ◽  
Free Abelian Group ◽  
Group Topology ◽  
Countably Compact
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