Free Topological Groups and the Projective Dimension of a Locally Compact Abelian Group

Mack, John; Morris, Sideny A.; Ordman, Edward T.

doi:10.2307/2038685

Cited by 35 publications

(42 citation statements)

References 5 publications

(9 reference statements)

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“…is topologically isomorphic to the free Abelian topological group A(Y ) [31]. Hence all compact subsets of A(Y, αX) ∼ = A(Y ) are ℵ 0 -monolithic by a result in [6].…”

Section: Compact Sets In Extensions Of Groupsmentioning

confidence: 94%

The three space problem in topological groups

Bruguera¹,

Tkachenko²

2006

Topology and its Applications

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We study compact, countably compact, pseudocompact, and functionally bounded sets in extensions of topological groups. A property P is said to be a three space property if, for every topological group G and a closed invariant subgroup N of G, the fact that both groups N and G/N have P implies that G also has P. It is shown that if all compact (countably compact) subsets of the groups N and G/N are metrizable, then G has the same property. However, the result cannot be extended to pseudocompact subsets, a counterexample exists under p = c. Another example shows that extensions of groups do not preserve the classes of realcompact, Dieudonné complete and μ-spaces: one can find a pseudocompact, non-compact Abelian topological group G and an infinite, closed, realcompact subgroup N of G such that G/N is compact and all functionally bounded subsets of N are finite. Several examples given in the article destroy a number of tempting conjectures about extensions of topological groups. (M. Bruguera), mich@xanum.uam.mx (M. Tkachenko).

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“…is topologically isomorphic to the free Abelian topological group A(Y ) [31]. Hence all compact subsets of A(Y, αX) ∼ = A(Y ) are ℵ 0 -monolithic by a result in [6].…”

Section: Compact Sets In Extensions Of Groupsmentioning

confidence: 94%

The three space problem in topological groups

Bruguera¹,

Tkachenko²

2006

Topology and its Applications

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“…Part (a) of the following result is essentially proved by Mack, Morris and Ordman (1973). (To be precise, they proved that T, = @; however, it is easily seen that similar techniques will prove that © = T P .)…”

Section: A 2 • • • a M ) = A L A 2 -• • A M )mentioning

confidence: 96%

Free topological groups

Borges

1977

J. Aust. Math. Soc.

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Let X be any Tychonoff space and /3X the Stone-Cech compactification of X. Let F(/3X) be the Graev free group of PX and let ©' be the subspace topology on the Graev group F(X). Our results demonstrate that this topology is useful and behaves extremely well; the behavior of the free topology still remains enigmatic.There are various applications, some of which clarify the free topology on F(X), while others improve various results recently published.Let X be any Tychonoff space and )3X the Stone-Cech compactification of X. Let F((1X) be the Graev free group of /3X and let ©' be the subgroup topology on the Graev group F(X). Our results demonstrate that this topology is useful and behaves extremely well (see Lemma 2.2 and Theorem Z.3); the behavior of the free topology still remains enigmatic.There are various applications, some of which clarify the free topology on F(X) (see Theorem 2.4), while others improve various results recently published by B. V. S. Thomas (see Propositions 3.1, 3.2 and 3.3).

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“…2 (B) For a compact Hausdorff space K , the free abelian topological group A(K ) is a hemicompact k-space, and a cobasis for the compact sets is given by the family of sets {η(K )+ n · · · +η(K ): n ∈ N} [24]. (C) For a compact Hausdorff space K , the dual group of A(K ) is topologically isomorphic to C(K , T); more precisely, [29].…”

Section: (B) There Exists a Gtg Neighborhood Of Zero U Such That G Ismentioning

confidence: 99%