Nöbeling spaces and pseudo-interiors of Menger compacta

Chigogidze, Alex; Kawamura, Kazuhiro; Tymchatyn, E. D.

doi:10.1016/0166-8641(96)00041-7

Cited by 15 publications

(3 citation statements)

References 26 publications

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“…It is known that the n-dimensional Nöbeling space ν 2n+1 n is a strongly A ω,n -universal, AN E(n)-space. The following three statements are proved in [5] as Proposition 3.6, Lemma 3.2 and Proposition 3.8, respectively. Definition 9.…”

Section: ) F I+1mentioning

confidence: 92%

Spaces of σ-finite linear measure

Stasyuk

Tymchatyn

2013

Colloq. Math.

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Spaces of finite n-dimensional Hausdorff measure are an important generalization of n-dimensional polyhedra. Continua of finite linear measure (also called continua of finite length) were first characterized by Eilenberg in 1938. It is well-known that the property of having finite linear measure is not preserved under finite unions of closed sets. Mauldin proved that if X is a compact metric space which is the union of finitely many closed sets each of which admits a σ-finite linear measure then X admits a σ-finite linear measure. We answer in the strongest possible way a 1989 question (private communication) of Mauldin. We prove that if a separable metric space is a countable union of closed subspaces each of which admits finite linear measure then it admits σ-finite linear measure. In particular, it can be embedded in the 1-dimensional Nöbeling space ν 3 1 so that the image has σ-finite linear measure with respect to the usual metric on ν 3 1 .

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Section: ) F I+1mentioning

confidence: 92%

Spaces of σ-finite linear measure

Stasyuk

Tymchatyn

2013

Colloq. Math.

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“…Since every compact Menger manifold (a manifold modeled on the Menger cube µ n for some n ≥ 1), as well as every 1-dimensional Peano continuum, admits small retractions to compact polyhedra, it was observed in [7,] that any such a space is Krasinkiewicz. Moreover, every Nöbeling manifold also admits small retractions to polyhedra, see [1]. So, by Proposition 3.3, we have: Proof.…”

Section: Some Properties Of Krasinkiewicz Spacesmentioning

confidence: 93%

Krasinkiewicz spaces and parametric Krasinkiewicz maps

Matsuhashi,

Valov

2008

Preprint

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We say that a metrizable space M is a Krasinkiewicz space if any map from a metrizable compactum X into M can be approximated by Krasinkiewicz maps (a map g : X → M is Krasinkiewicz provided every continuum in X is either contained in a fiber of g or contains a component of a fiber of g). In this paper we establish the following property of Krasinkiewicz spaces: Let f : X → Y be a perfect map between metrizable spaces and M a Krasinkiewicz complete AN R-space. If Y is a countable union of closed finite-dimensional subsets, then the function space C(X, M ) with the source limitation topology contains a dense G δ -subset of maps g such that all restrictions g|f −1 (y), y ∈ Y , are Krasinkiewicz maps. The same conclusion remains true if M is homeomorphic to a closed convex subset of a Banach space and X is a C-space.

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“…First of all we note that the (n + 1)-dimensional universal Menger compactum µ n+1 also admits [9] a Z-skeletoid Σ n+1 such that its complement ν n+1 = µ n+1 − Σ n+1 is homeomorphic [10], [7,Theorem 5.5.5] to the (n + 1)-dimensional universal Nöbeling space N 2n+3 n+1 . Now we proceed as above.…”

Section: Resultsmentioning

confidence: 99%