scholarly journals Local invariance of free topological groups

1986 ◽  
Vol 29 (1) ◽  
pp. 1-5 ◽  
Author(s):  
M. S. Khan ◽  
Sidney A. Morris ◽  
Peter Nickolas
Keyword(s):  
Topological Group ◽  
Hausdorff Space ◽  
Convergent Sequence ◽  
Discrete Topology ◽  
Group Topology ◽  
Topological Groups ◽  
Completely Regular ◽  
Free Topological Group ◽  
Hausdorff Group ◽  
Totally Disconnected

In 1948, M. I. Graev [2] proved that the free topological group on a completely regular Hausdorff space is Hausdorff, by showing that the free group admits a certain locally invariant Hausdorff group topology. It is natural to ask if Graev's locally invariant topology is the free topological group topology. If X has the discrete topology, the answer is clearly in the affirmative. In 1973, Morris-Thompson [6] showed that if X is not totally disconnected then the answer is negative. Nickolas [7] showed that this is also the case if X has any (non-trivial) convergent sequence (for example, if X is any non-discrete metric space). Recently, Fay and Smith Thomas handled the case when X has a completely regular Hausdorff quotient space which has an infinite compact subspace (or more particularly a non-trivial convergent sequence).(Fay-Smith Thomas observe that their class of spaces includes some but not all those dealt with by Morris-Thompson.)

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 A remark on free topological groups with no small subgroups

1974 ◽  
Vol 18 (4) ◽  
pp. 482-484 ◽  
Author(s):  
H. B. Thompson
Keyword(s):  
Topological Group ◽  
Free Group ◽  
Regular Space ◽  
Group Topology ◽  
Topological Properties ◽  
Topological Groups ◽  
Abstract Group ◽  
Completely Regular ◽  
Metric Topology ◽  
Totally Disconnected

For a completely regular space X let G(X) be the Graev free topological group on X. While proving G(X) exists for completely regular spaces X, Graev showed that every pseudo-metric on X can be extended to a two-sided invariant pseudo-metric on the abstract group G(X). The free group topology on G(X) is usually strictly finer than this pseudo-metric topology. In particular this is the case when X is not totally disconnected (see Morris and Thompson [7]). It is of interest to know when G(X) has no small subgroups (see Morris [5]). Morris and Thompson [6] showed that this is the case if and only if X admits a continuous metric. The proof relied on properties of the free group topology and it is natural to ask if G(X) with its pseudo-metric topology has no small subgroups when and only when X admits a continuous metric. We show that this is the case. Topological properties of G(X) associated with the pseudo-metric topology have recently been studied by Joiner [3] and Abels [1].

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 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).

scholarly journals VARIETIES OF ABELIAN TOPOLOGICAL GROUPS AND SCATTERED SPACES

2008 ◽  
Vol 78 (3) ◽  
pp. 487-495 ◽  
Author(s):  
CAROLYN E. MCPHAIL ◽  
SIDNEY A. MORRIS
Keyword(s):  
Topological Group ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Topological Groups ◽  
Abelian Topological Group ◽  
Free Abelian Topological Group ◽  
Scattered Spaces

AbstractThe variety of topological groups generated by the class of all abelian kω-groups has been shown to equal the variety of topological groups generated by the free abelian topological group on [0, 1]. In this paper it is proved that the free abelian topological group on a compact Hausdorff space X generates the same variety if and only if X is not scattered.

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.

scholarly journals Free products of topological groups

1971 ◽  
Vol 4 (1) ◽  
pp. 17-29 ◽  
Author(s):  
Sidney A. Morris
Keyword(s):  
Topological Group ◽  
Free Product ◽  
Quotient Group ◽  
Closed Subgroup ◽  
Topological Product ◽  
Almost Periodic ◽  
Topological Groups ◽  
Free Products ◽  
Free Topological Group ◽  
Product Of Groups

In this note the notion of a free topological product Gα of a set {Gα} of topological groups is introduced. It is shown that it always exists, is unique and is algebraically isomorphic to the usual free product of the underlying groups. Further if each Gα is Hausdorff, then Gα is Hausdorff and each Gα is a closed subgroup. Also Gα is a free topological group (respectively, maximally almost periodic) if each Gα is. This notion is then combined with the theory of varieties of topological groups developed by the author. For a variety of topological groups, the -product of groups in is defined. It is shown that the -product, Gα of any set {Gα} of groups in exists, is unique and is algebraically isomorphic to the usual varietal product. It is noted that the -product of Hausdorff groups is not necessarily Hausdorff, but is if is abelian. Each Gα is a quotient group of Gα. It is proved that the -product of free topological groups of and projective topological groups of are of the same type. Finally it is shown that Gα is connected if and only if each Gα is connected.

scholarly journals Varieties of topological groups and left adjoint functors

1973 ◽  
Vol 16 (2) ◽  
pp. 220-227 ◽  
Author(s):  
Sidney A. Morris
Keyword(s):  
Topological Group ◽  
Topological Space ◽  
Left Adjoint ◽  
Topological Groups ◽  
Adjoint Functors ◽  
Free Topological Group

In [6] and [2] Markov and Graev introduced their respective concepts of a free topological group. Graev's concept is more general in the sense that every Markov free topological group is a Graev free topological group. In fact, if FG(X) is the Graev free topological group on a topological space X, then it is the Markov free topological group FM(Y) on some space Y if and only if X is disconnected. This, however, does not say how FG(X) and FM(X) are related.

scholarly journals LECs, Local Mixers, Topological Groups and Special Products

1988 ◽  
Vol 44 (2) ◽  
pp. 252-258 ◽  
Author(s):  
Carlos R. Borges
Keyword(s):  
Topological Group ◽  
Topological Groups ◽  
Symmetric Products ◽  
Free Topological Group ◽  
Reduced Products ◽  
Contractible Spaces ◽  
Homeomorphism Groups

AbstractWe prove that every (locally) contractible topological group is (L)EC and apply these results to homeomorphism groups, free topological groups, reduced products and symmetric products. Our main results are: The free topological group of a θ-contractible space is equiconnected. A paracompact and weakly locally contractible space is locally equiconnected if and only if it has a local mixer. There exist compact metric contractible spaces X whose reduced (symmetric) products are not retracts of the Graev free topological groups F(X) (A(X)) (thus correcting results we published ibidem).

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