Local invariance of free topological groups

Khan, M. S.; Morris, Sidney A.; Nickolas, Peter

doi:10.1017/s001309150001734x

Cited by 2 publications

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References 6 publications

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“…It is well known that Graev's topology is only equal to the free topology in somewhat pathological cases [9,19,22]. In the abelian case, a parallel construction was also outlined by Graev, and in this case Graev's topology is always the free topology (see [13,11,20]; cf.…”

Section: The Character Of Free Abelian Topological Groupsmentioning

confidence: 99%

The character of free topological groups I

Nickolas

Tkachenko

2005

Appl.Gen.Topol.

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Abstract.A systematic analysis is made of the character of the free and free abelian topological groups on uniform spaces and on topological spaces. In the case of the free abelian topological group on a uniform space, expressions are given for the character in terms of simple cardinal invariants of the family of uniformly continuous pseudometrics of the given uniform space and of the uniformity itself. From these results, others follow on the basis of various topological assumptions. Amongst these: (i) if X is a compact Hausdorff space, then the character of the free abelian topological group on X lies between w(X) and w(X) ℵ 0 , where w(X) denotes the weight of X; (ii) if the Tychonoff space X is not a P -space, then the character of the free abelian topological group is bounded below by the "small cardinal" d; and (iii) if X is an infinite compact metrizable space, then the character is precisely d. In the non-abelian case, we show that the character of the free abelian topological group is always less than or equal to that of the corresponding free topological group, but the inequality is in general strict. It is also shown that the characters of the free abelian and the free topological groups are equal whenever the given uniform space is ω-narrow. A sequel to this paper analyses more closely the cases of the free and free abelian topological groups on compact Hausdorff spaces and metrizable spaces.2000 AMS Classification: Primary 22A05, 54H11, 54A25; Secondary 54D30, 54D45.

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Section: The Character Of Free Abelian Topological Groupsmentioning

confidence: 99%

The character of free topological groups I

Nickolas

Tkachenko

2005

Appl.Gen.Topol.

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“…Graev's proof in [5] of the fact that the free topological group on a Tychonoff space X is Hausdorff proceeds by a construction which extends each continuous pseudometric on X to an invariant pseudometric on the underlying abstract free group F a (X), and an argument which shows that the group topology induced by all the extensions (referred to as Graev's topology) is Hausdorff and weaker than the free topology. It is well known that Graev's topology is only equal to the free topology in somewhat pathological cases [9,19,22]. In the abelian case, a parallel construction was also outlined by Graev, and in this case Graev's topology is always the free topology (see [13,11,20]; cf.…”

Section: The Character Of Free Abelian Topological Groupsmentioning

confidence: 99%

Free topological groups over metrizable spaces

Arhangel'skiı̆

Okunev

Pestov

1989

Topology and its Applications

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A systematic analysis is made of the character of the free and free abelian topological groups on uniform spaces and on topological spaces. In the case of the free abelian topological group on a uniform space, expressions are given for the character in terms of simple cardinal invariants of the family of uniformly continuous pseudometrics of the given uniform space and of the uniformity itself. From these results, others follow on the basis of various topological assumptions. Amongst these: (i) if X is a compact Hausdorff space, then the character of the free abelian topological group on X lies between w(X) and w(X) ℵ 0 , where w(X) denotes the weight of X; (ii) if the Tychonoff space X is not a P-space, then the character of the free abelian topological group is bounded below by the "small cardinal" d; and (iii) if X is an infinite compact metrizable space, then the character is precisely d. In the non-abelian case, we show that the character of the free abelian topological group is always less than or equal to that of the corresponding free topological group, but the inequality is in general strict. It is also shown that the characters of the free abelian and the free topological groups are equal whenever the given uniform space is ω-narrow. A sequel to this paper analyses more closely the cases of the free and free abelian topological groups on compact Hausdorff spaces and metrizable spaces.

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Local invariance of free topological groups

Cited by 2 publications

References 6 publications

The character of free topological groups I

The character of free topological groups I

Free topological groups over metrizable spaces

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