Applications of the Stone-Čech compactification to free topological groups

Hardy, J. P. L.; Morris, Sidney A.; Thompson, H. B.

doi:10.1090/s0002-9939-1976-0424994-x

Cited by 6 publications

(7 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To show that the groups F(X n ) occur as subgroups of F(X) for suitable X, we must first find in F(X) a copy of X" which is algebraically a free basis for the subgroup it generates. It is shown in [3] and [16] that for any space X, the map <f>: (x u x 2 ,..., x n ) i-> x t x 2 2 x 3 * ... x,, 2 "' 1 is a homeomorphism of X" into F(X). It is easy to see however that under no mapping of the form 0 :(x u x 2 ,..., x n )\-*x l kl x 2 k2 ... jc n fcn (for fixed integers {fcj) is <p(X n ) a free basis for gp($(X")).…”

Section: Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Subgroups of the Free Topological Group on [0, 1]

Nickolas

1976

Journal of the London Mathematical Society

View full text Add to dashboard Cite

Section: Resultsmentioning

confidence: 99%

“…If (1) Conversely, suppose that (2) holds, and denote by / the copy of [0,1] which generates the subgroup in question. By an argument in the spirit of those in [3], we see that for some n (which we choose to be minimal), I c F n ( 7). (In fact, let (f>: Y -* /?…”

Section: Andmentioning

confidence: 99%

Subgroups of the Free Topological Group on [0, 1]

Nickolas

1976

Journal of the London Mathematical Society

View full text Add to dashboard Cite

“…The next result from [1,8,7] provides important information about the topological structure of the subspaces F n (X) C F(X) and A n (X) C A(X) of elements of length precisely n. …”

Section: Tie Family {O(s) : S E {Uxy} Is a Local Base At The Neutramentioning

confidence: 99%

Local compactness in free topological groups

Nickolas¹,

Tkachenko²

2003

Bull. Austral. Math. Soc.

View full text Add to dashboard Cite

We show that the subspace A n (X) of the free Abelian topological group A(X) on a Tychonoff space X is locally compact for each n € u> if and only if A2(X) is locally compact if and only if ^( X ) is locally compact if and only if X is the topological sum of a compact space and a discrete space. It is also proved that the subspace F n (X) of the free topological group F(X) is locally compact for each n G w if and only if Fn(X) is locally compact if and only if F n (X) has pointwise countable type for each n e w i f and only if F^{X) has pointwise countable type if and only if X is either compact or discrete, thus refining a result by Pestov and Yamada. We further show that A n {X) has pointwise countable type for each n € w if and only if A2(X) has pointwise countable type if and only if •FM-Y) has pointwise countable type if and only if there exists a compact set C of countable character in X such that the complement X \ C is discrete. Finally, we show that Fi(X) is locally compact if and only if F3(X) is locally compact, and that Fz(X) has pointwise countable type if and only if F$(X) has pointwise countable type.

show abstract

“…This can be proved most easily by observing that it is true when X is compact, since FA(X) is then a fcw-space with ^-decomposition FA(X) = \\FAn(X). The noncompact case can then be established using Stone-Cech compactification in the manner described in [7]. )…”

Section: Resultsmentioning

confidence: 99%

Sequential conditions and free products of topological groups

Morris¹,

Thompson²

1988

Proc. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

ABSTRACT. If A and B are nontrivial topological groups, not both discrete, such that their free product A 11 B is a sequential space, then it is sequential of order oji.1. Introduction.In [15], Ordman and Smith-Thomas prove that if the free topological group on a nondiscrete space is sequential then it is sequential of order uji. In particular, this implies that free topological groups are not metrizable or even Fréchet spaces. Our main result is the analogue of this for free products of topological groups. More precisely we prove that if A and B are nontrivial topological groups not both discrete and their free product A II B is sequential, then it is sequential of order ui. This result is then extended to some amalgamated free products. Our theorem includes, as a special case, the result of [10]. En route we extend the Ordman and Smith-Thomas result to a number of other topologies on a free group including the Graev topology which is the finest locally invariant group topology. (See [12].) It should be mentioned, also, that O.dman and SmithThomas show that the condition of AII B being sequential is satisfied whenever A and B are sequential fc^-spaces.

show abstract

Applications of the Stone-Čech compactification to free topological groups

Cited by 6 publications

References 11 publications

Subgroups of the Free Topological Group on [0, 1]

Subgroups of the Free Topological Group on [0, 1]

Local compactness in free topological groups

Sequential conditions and free products of topological groups

Contact Info

Product

Resources

About