On extending homeomorphisms on zero-dimensional spaces

Pollard, Jean

doi:10.4064/fm-67-1-39-48

Cited by 11 publications

(11 citation statements)

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“…As for (3), if A ∈ 2 X satisfies A S then, by S = D, we can find a sequence B i in 2 X converging to A and satisfying B i D and |B i | < ℵ 0 . Thus A ∈ S. Conversely, if A − S = ∅ then such a sequence does not exist and hence A / ∈ S. 2 Lemma 2 (Ryll and Nardzewski; see [1,Corollary 2], also see J. Pollard's paper [5]). Let C, C be Cantor sets and B.…”

Section: Candidates For 2 Xmentioning

confidence: 93%

The topological types of hyperspaces of 0-dimensional compacta

Oka

2005

Topology and its Applications

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Let X be a 0-dimensional compactum (= compact metric space) and let 2 X denote the hyperspace of X, the set of non-empty closed subsets of X with the Hausdorff metric. We shall determine all the topological types of the hyperspaces 2 X . They are only finitely many in the non-trivial case where the scattered part of X contains infinitely many points.  2004 Published by Elsevier B.V. MSC: 54B20; 54E45Keywords: Hyperspace; 0-dimensional compactum IntroductionLet X be a 0-dimensional compactum (= compact metric space) and let 2 X denote the set of non-empty closed subsets of X with the Hausdorff metric. As well known, 2 X is a 0-dimensional compactum again, and if X is a Cantor set then so is 2 X . By the CantorBendixson theorem, X is uniquely decomposed as X = K ∪ S with K a closed set of X having no isolated points and S a scattered space. The kernel K, if not empty, is homeomorphic to the Cantor set. We also use the symbol D to denote the isolated points of X. Note that S D, with D the closure of D.In the special case where S is dense in X, the structure of 2 X was studied by Pelczyński and found to be uniquely determined if S contains infinitely many points ([3]; also see [2]).

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Section: Candidates For 2 Xmentioning

confidence: 93%

The topological types of hyperspaces of 0-dimensional compacta

Oka

2005

Topology and its Applications

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“…Finally, since U Y L and U 1 Y L 1 are copies of the irrationals, a theorem of Pollard [14] provides an extension of f to a homeomorphism g :…”

Section: 2mentioning

confidence: 99%

“…We can now use the claim to define a generalized Hurewicz system pU 1 s q sPN ăN , pL 1 s q sPN ăN of subsets of G 1 together with homeomorphisms h s satisfying conditions (10)- (14) with (12) replaced by the requirements that µpL 1 s q ą 0 and h s |L s : L sÑ L 1 s preserves measure up to a positive constant factor.…”

Section: 2mentioning

confidence: 99%

On Borel maps, calibrated ${\ensuremath {\sigma }}$-ideals, and homogeneity

Pol

Zakrzewski

2018

Trans. Amer. Math. Soc.

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Let µ be a Borel measure on a compactum X. The main objects in this paper are σ-ideals Ipdimq, J 0 pµq, J f pµq of Borel sets in X that can be covered by countably many compacta which are finite-dimensional, or of µ-measure null, or of finite µmeasure, respectively. Answering a question of J. Zapletal, we shall show that for the Hilbert cube, the σ-ideal Ipdimq is not homogeneous in a strong way. We shall also show that in some natural instances of measures µ with non-homogeneous σ-ideals J 0 pµq or J f pµq, the completions of the quotient Boolean algebras BorelpXq{J 0 pµq or BorelpXq{J f pµq may be homogeneous.We discuss the topic in a more general setting, involving calibrated σ-ideals.2010 Mathematics Subject Classification. 03E15, 54H05, 28A78, 54F45.

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“…Objects similar to those defined above were used by Knaster and Reichbach in [14] to prove some theorems on extensions of homeomorphisms; their technique has afterwards been used by various authors, e.g. Pollard [19], Ravdin [20], and Gutek [10].…”

Section: Definitionmentioning

confidence: 99%