Preservation of the Borel class under open-LC functions

Ostrovsky, Alexey

doi:10.4064/fm213-2-4

Cited by 4 publications

(2 citation statements)

References 6 publications

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“…The present paper continues the series of publications about decomposibility of Borel functions [6], [8] -see also [2], [3] where functions of such type are the main subject.…”

Section: Introductionmentioning

confidence: 57%

Open-constructible functions

Ostrovsky

2014

Topology and its Applications

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Abstract. Let f be a continuous function between subspaces X, Y of the Cantor set C. We prove that:if f is one-to-one and maps open sets into resolvable, then f is a piecewise homeomorphism and if f maps discrete subsets into resolvable, then f is piecewise open. IntroductionThe present paper continues the series of publications about decomposibility of Borel functions [6], [8] -see also [2], [3] where functions of such type are the main subject.A subset E of a topological space X is resolvable if for each nonempty closed in X subset F we have Note, that using standard sets H, we obtain the proof for open-resolvable functions simpler than for open-constructible.Analogously, using H-sets we can easy extend the proof for closed-constructible functions in [6] to the case of closed-resolvable.Standard set H will be used in the proof of the following theorem:Theorem 2. If a continuous function f : X → Y between X, Y ⊂ C maps discrete subsets in X onto resolvable, then f is piecewise open.2000 Mathematics Subject Classification. Primary 54E40, 03E15, 26A21; Secondary 54H05, 28A05, 03G05.

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“…The present paper continues the series of publications about decomposibility of Borel functions [6], [8] -see also [2], [3] where functions of such type are the main subject.…”

Section: Introductionmentioning

confidence: 57%

Open-constructible functions

Ostrovsky

2014

Topology and its Applications

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“…The author has limited his first work to only one question about the preservation of completeness, since it was not clear whether such a decomposition was possible. It is known, however, that if a decomposition is possible, it leads to preservation of even other Borel classes [9]. Recently, this question was fully and affirmatively resolved by Gao and Kleinfield [1], Holický and Pol [3].…”

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confidence: 98%

Closed-constructible functions are piecewise closed

Ostrovsky¹

2013

Topology and its Applications

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Dedicated to the memory of topologist and friend D. Ranchin MSC: primary 54C10, 28A05, 03E15, 54E40, 26A21A subset of a topological space X is constructible if it belongs to the smallest algebra of subsets that contains all open subsets of X. We prove that if a continuous function f : X → Y between separable metrizable spaces maps closed subsets of X onto constructible sets of Y , then X admits a countable closed cover C such that for each C ∈ C the restriction f |C is closed.The present paper continues the series of publications about decomposibility of Borel functions 1 f between separable metrizable spaces into a finite or countable number of open, closed and continuous functions.In the first publication [7] we have raised the question about preservation of complete metrizability by some simplest Borel functions. The author has limited his first work to only one question about the preservation of completeness, since it was not clear whether such a decomposition was possible. It is known, however, that if a decomposition is possible, it leads to preservation of even other Borel classes [9]. Recently, this question was fully and affirmatively resolved by Gao and Kleinfield [1], Holický and Pol [3].In the following publications [8,10] we have obtained such a decomposition into open, closed and continuous functions in the case of simplest Borel functions.In the present paper we give an affirmative solution for the first case when images of closed sets 2 are unrestricted combinations of closed and open sets.

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An alternative approach to the decomposition of functions

Ostrovsky¹

2012

Topology and its Applications

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Preservation of the Borel class under open-LC functions

Cited by 4 publications

References 6 publications

Open-constructible functions

Open-constructible functions

Closed-constructible functions are piecewise closed

An alternative approach to the decomposition of functions

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