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 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 On orderability of topological groups

1985 ◽  
Vol 8 (4) ◽  
pp. 747-754
Author(s):  
G. Rangan
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Group Structure ◽  
Locally Compact Abelian Group ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Topological Groups ◽  
Locally Compact ◽  
Necessary And Sufficient ◽  
Totally Disconnected

A necessary and sufficient condition for a topological group whose topology can be induced by a total order compatible with the group structure is given and such groups are called ordered or orderable topological groups. A separable totally disconnected ordered topological group is proved to be non-archimedean metrizable while the converse is shown to be false by means of an example. A necessary and sufficient condition for a no-totally disconnected locally compact abelian group to be orderable is also given.

scholarly journals On group uniformities on the square of a space and extending pseudometrics

1995 ◽  
Vol 51 (2) ◽  
pp. 309-335 ◽  
Author(s):  
Michael G. Tkačnko
Keyword(s):  
Topological Group ◽  
Topological Groups ◽  
Free Topological Group

We give some conditions under which, for a given pair (d1, d2) of continuous pseudometrics respectively on X and X3, there exists a continuous semi-norm N on the free topological group F(X) such that N(x · y−1) = d1(x, y) and N(x · y · t−1 · z−1) ≥ d2((x, y), (z, t)) for all x, y, z, t ∈ X. The “extension” results are applied to characterise thin subsets of free topological groups and obtain some relationships between natural uniformities on X2 and those induced by the group uniformities *V, V* and *V* of F(X).

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.

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 A Schreier theorem for free topological groups

1975 ◽  
Vol 13 (1) ◽  
pp. 121-127 ◽  
Author(s):  
Peter Nickolas
Keyword(s):  
Topological Group ◽  
Closed Subgroup ◽  
Open Subgroup ◽  
Necessary Condition ◽  
Topological Groups ◽  
Free Topological Group

M.I.Graev has shown that subgroups of free topological groups need not be free. Brown and Hardy, however, have proved that any open subgroup of the free topological group on a kw-space is again a free topological group: indeed, this is true for any closed subgroup for which a Schreier transversal can be chosen continuously. This note provides a proof of this result more direct than that of Brown and Hardy. An example is also given to show that the condition stated in the theorem is not a necessary condition for freeness of a subgroup. Finally, a sharpened version of a particular case of the theorem is obtained, and is applied to the preceding example.

Varieties of topological groups

1969 ◽  
Vol 1 (2) ◽  
pp. 145-160 ◽  
Author(s):  
Sidney A. Morris
Keyword(s):  
Topological Group ◽  
Topological Space ◽  
Free Group ◽  
Quotient Group ◽  
Sufficient Conditions ◽  
Algebraic Theory ◽  
Topological Groups ◽  
Free Basis ◽  
Free Topological Group ◽  
Necessary And Sufficient

We introduce the concept of a variety of topological groups and of a free topological group F(X, ) of on a topological space X as generalizations of the analogous concepts in the theory of varieties of groups. Necessary and sufficient conditions for F(X, ) to exist are given and uniqueness is proved. We say the topological group FM,(X) is moderately free on X if its topology is maximal and it is algebraically free with X as a free basis. We show that FM(X) is a free topological group of the variety it generates and that if FM(X) is in then it is topologically isomorphic to a quotient group of F(X, ). It is also shown how well known results on free (free abelian) topological groups can be deduced. In the algebraic theory there are various equivalents of a free group of a variety. We examine the relationships between the topological analogues of these. In the appendix a result similar to the Stone-Čech compactification is proved.

SUBSPACES OF THE FREE TOPOLOGICAL VECTOR SPACE ON THE UNIT INTERVAL

2017 ◽  
Vol 97 (1) ◽  
pp. 110-118 ◽  
Author(s):  
SAAK S. GABRIYELYAN ◽  
SIDNEY A. MORRIS
Keyword(s):  
Topological Group ◽  
Vector Space ◽  
Topological Vector Space ◽  
Unit Interval ◽  
Vector Subspace ◽  
Locally Compact ◽  
Tychonoff Space ◽  
Abelian Topological Group ◽  
Free Abelian Topological Group ◽  
Usual Topology

For a Tychonoff space $X$, let $\mathbb{V}(X)$ be the free topological vector space over $X$, $A(X)$ the free abelian topological group over $X$ and $\mathbb{I}$ the unit interval with its usual topology. It is proved here that if $X$ is a subspace of $\mathbb{I}$, then the following are equivalent: $\mathbb{V}(X)$ can be embedded in $\mathbb{V}(\mathbb{I})$ as a topological vector subspace; $A(X)$ can be embedded in $A(\mathbb{I})$ as a topological subgroup; $X$ is locally compact.

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.

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