Applications of the Stone-Cech Compactification to Free Topological Groups

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

doi:10.2307/2041864

Cited by 12 publications

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“…If X is compact, then clearly it is a closed subset of L p (X), L(X), V(X), F(X), A(X) and B(X). Indeed by the Stone-Cech compactification argument in [10], for any Tychonoff space X, the space X is a closed subset of each of these. (For example, if X is any Tychonoff space and φ : X → βX is the canonical one-to-one continuous map into its Stone-Cech compactification, there is a continuous linear operator Φ from V(X) to V(βX) which extends φ.…”

Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

“…Joiner [12] gave a useful description of the topology of the words of (reduced) length n in the free topological group and the free abelian topological group on a Tychonoff space X. A simpler proof of this result was given by Hardy, Morris and Thompson in [10] using Stone-Cech compactifications. Unknown to Joiner, Hardy, Morris and Thompson, A.V.…”

Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

“…Next we generalise Proposition 1.16 from the compact case to a Tychonoff space X by applying the standard Stone-Cech compactification argument of [10]. Theorem 1.17.…”

Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

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Embeddings of Free Topological Vector Spaces

Leiderman

Morris

2019

Bull. Aust. Math. Soc.

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It is proved that the free topological vector space $\mathbb{V}([0,1])$ contains an isomorphic copy of the free topological vector space $\mathbb{V}([0,1]^{n})$ for every finite-dimensional cube $[0,1]^{n}$, thereby answering an open question in the literature. We show that this result cannot be extended from the closed unit interval $[0,1]$ to general metrisable spaces. Indeed, we prove that the free topological vector space $\mathbb{V}(X)$ does not even have a vector subspace isomorphic as a topological vector space to $\mathbb{V}(X\oplus X)$, where $X$ is a Cook continuum, which is a one-dimensional compact metric space. This is also shown to be the case for a rigid Bernstein set, which is a zero-dimensional subspace of the real line.

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Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

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Embeddings of Free Topological Vector Spaces

Leiderman

Morris

2019

Bull. Aust. Math. Soc.

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“…2 Given a space X and a natural number n, we denote by A n (X) the subspace of the free Abelian topological group A(X) which consists of all words of reduced length n with respect to the basis X. It is well known that A n (X) is closed in A(X) for each n ∈ ω (see [25] or [6]). We need the following simple fact.…”

Section: Lemma 47 Let E Be An Infinite Subset Of ω and γ Be A Subfamentioning

confidence: 99%

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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“…The work of [5] and [7 2] shows that if X is a fe-space such that the cartesian product X x X is not a fc-space, then the free topological group F(X) is not a fe-space. In particular, then, if X is Dowker's CV-complex [2] the free topological group F(X) is a priori not a CVcomplex, since it is not even a fe-space.…”

Section: Introductionmentioning

confidence: 99%