A Fake Topological Hilbert Space

Anderson, R. D.; Curtis, D. W.; Mill, J. Van

doi:10.2307/1998961

Cited by 21 publications

(24 citation statements)

References 4 publications

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“…Let Y be as in Theorem 3.13 (2). Observe that Y has the SP(see §1) and hence has a dense Polish subspace by Corollary 3.2(2) (this can also be seen directly of course).…”

Section: Example 312mentioning

confidence: 81%

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Analytic groups and pushing small sets apart

Mill

2009

Trans. Amer. Math. Soc.

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Abstract. We say that a space X has the separation property provided that if A and B are subsets of X with A countable and B first category, then there is a homeomorphism f : X → X such that f (A) ∩ B = ∅. We prove that a Borel space with this property is Polish. Our main result is that if the homeomorphisms needed in the separation property for the space X come from the homeomorphisms given by an action of an analytic group, then X is Polish. Several examples are also presented.

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“…Let Y be as in Theorem 3.13 (2). Observe that Y has the SP(see §1) and hence has a dense Polish subspace by Corollary 3.2(2) (this can also be seen directly of course).…”

Section: Example 312mentioning

confidence: 81%

“…But then f (U ) belongs to U since f is a homeomorphism, and f (U ) ⊆ A, which evidently is a contradiction. We now prove (2). Let G ⊆ X be a dense Polish subspace in X.…”

Section: Terminologymentioning

confidence: 94%

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Analytic groups and pushing small sets apart

Mill

2009

Trans. Amer. Math. Soc.

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“…Moreover, modifying the argument of [ACvM,Lemma 3.6], one sees that both π 1 and π 2 are monotone. To see that π 1 is monotone fix (x, t) ∈ Q .…”

Section: The Space Ymentioning

confidence: 99%

A Polish AR-Space with no Nontrivial Isotopy

Dobrowolski

2008

Bull. Polish Acad. Sci. Math.

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“…The separable Hilbert space l2 satisfies the discrete oo-cells property, usually referred to as the discrete approximation property [To2], and the author has constructed for each « e iV U {0}, examples of complete (« + l)-dimensional AR's that satisfy the discrete «-cells property [Bo,]. For further results about discrete cells properties, see [To2, ACM,Cu2,Bo,,Bo2].…”

Section: Corollarymentioning

confidence: 99%

Dense Embeddings of Sigma-Compact, Nowhere Locally Compact Metric Spaces

Bowers¹

1985

Proceedings of the American Mathematical Society

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Abstract.It is proved that a connected complete separable ANR Z that satisfies the discrete n-cells property admits dense embeddings of every «-dimensional o-compact, nowhere locally compact metric space X(n e N U {0, oo}). More generally, the collection of dense embeddings forms a dense Gs-subset of the collection of dense maps of X into Z. In particular, the collection of dense embeddings of an arbitrary o-compact, nowhere locally compact metric space into Hubert space forms such a dense Cs-subset. This generalizes and extends a result of Curtis [Cu,]. 0. Introduction. In [Cu,], D. W. Curtis constructs dense embeddings of a-compact, nowhere locally compact metric spaces into the separable Hilbert space l2. In this paper, we extend and generalize this result in two distinct ways: first, we show that any dense map of a a-compact, nowhere locally compact space into Hilbert space is strongly approximable by dense embeddings, and second, we extend this result to finite-dimensional analogs of Hilbert space. Specifically, we prove Theorem. Let Z be a complete separable ANR that satisfies the discrete n-cells property for some w e ÍV U {0, oo} and let X be a o-compact, nowhere locally compact metric space of dimension at most n. Then for every map f: X -» Z such that f(X) is dense in Z and for every open cover <% of Z, there exists an embedding g: X -> Z such that g(X) is dense in Z and g is °U-close to f.We actually prove a slightly stronger version of the Theorem. We prove that the collection of dense embeddings of X into Z forms a dense C7s-subset of the collection of dense maps of X into Z topologized by the limitation topology. Corollary.Let n and Z be as in the Theorem with the added assumption that Z is connected. Then Z admits a dense embedding of every a-compact, nowhere locally compact metric space X of dimension at most n.Proof. It suffices to show that there is at least one map X -> Z whose image is dense in Z. Let R denote the following subset of the plane:R is a separable AR and since X is noncompact, there is a map of X onto a dense subset of R. (Let {x,} be a countable closed discrete subset of X and {/-,.} a

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A Fake Topological Hilbert Space

Cited by 21 publications

References 4 publications

Analytic groups and pushing small sets apart

Analytic groups and pushing small sets apart

A Polish AR-Space with no Nontrivial Isotopy

Dense Embeddings of Sigma-Compact, Nowhere Locally Compact Metric Spaces

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