The free topological group over the rationals

Fay, Temple H.; Ordman, Edward T.; Thomas, Barbara V. Smith

doi:10.1016/0016-660x(79)90027-8

Cited by 28 publications

(10 citation statements)

References 11 publications

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“…By a Theorem of Graev [14] G(βX) is a hemicompact k-space so that G(βX) is the Dieudonné completion (and also the Hewitt realcompactification) of G(X). [13, 8E] and, consequently, there exists G n (βX) such that cl G(βX) B ⊂ G n (βX) [12]. Since each G n (X) is closed in G(X), we have just proved that B ⊂ G n (X) so that G(X) is hemibounded.…”

Section: B F -Groupsmentioning

confidence: 78%

Dieudonné completion and bf-group actions

González

Sanchis

2006

Topology and its Applications

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If G(X) denotes either the free topological group or the free Abelian topological group over a topological space X, we prove that n i=1 G(X i ) is a hemibounded b f -group whenever each X i is a pseudocompact space (which provides a new way to generate this kind of topological groups), and we show that the equality μ(holds whenever X is a hemibounded b f -space (where μY stands for the Dieudonné completion of Y ). By means of the Dieudonné completion we prove that every pseudocompact space X is G-Tychonoff whenever G is a b f -group and that the maximal G-compactification of X coincides with βX. We apply this result to obtain a partial version for G-spaces of Glicksberg's theorem on pseudocompactness and we analyze when the maximal G-compactification of a G-space X coincides with the Stone-Čech compactification of X in the case when G is a metrizable group.

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Section: B F -Groupsmentioning

confidence: 78%

Dieudonné completion and bf-group actions

González

Sanchis

2006

Topology and its Applications

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“…The above theorem does not admit any noticeable further generalisation, apart from some openly pathological cases, such as the spaces X where every Gs set is open (the author, unpublished, 1981). In fact, it was shown in [36] that the mapping is is not quotient even for X = <Q>. Answering both questions raised in that paper, the author has proved the following result [114,117,120].…”

Section: Structure Of Free Topological Groupsmentioning

confidence: 94%

“…A very substantial body of results concerning the structure of free topological groups over k^ spaces have been deduced (mostly by Australian and American mathematicians) from Theorem 2.1 [23,24,36,48,67,68,69,70,71,72,73,95,96,105,106,107,112].…”

Section: Structure Of Free Topological Groupsmentioning

confidence: 99%

Universal arrows to forgetful functors from categories of topological algebra

Pestov¹

1993

Bull. Austral. Math. Soc.

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We survey the present trends in theory of universal arrows to forgetful functors from various categories of topological algebra and functional analysis to categories of topology and topological algebra. Among them are free topological groups, free locally convex spaces, free Banach-Lie algebras, and more. An accent is put on the relationship of those constructions with other areas of mathematics and their possible applications. A number of open problems is discussed; some of them belong to universal arrow theory, and other may become amenable to the methods of this theory.

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“…The following two problems have been extensively studied and proven very difficult for free and free Abelian topological groups. Results related to the inductive limit topology were mentioned in Section 3.1, and results related to the natural multiplication maps being quotient can be found, e.g., in [25,34,[71][72][73].…”

Section: A Few Open Problemsmentioning

confidence: 99%