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1
Content available A note on the hyperspace of finite subsets
Tuyen Luong Quoc, Tuyen Ong Van
Fasciculi Mathematici
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2021
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nr 65
67--73
EN
In this paper, we study the relation between a space X satisfying certain generalized metric properties and its hyperspace of finite subsets F(X) satisfying the same properties. We prove that if F(X) is a stric B0-space then so is X. However, there exists a stric B0-space X such that Fn(X) is not a stric B0-space for each n ≥ 2, hence F(X) is not a stric B0-space. Moreover, we prove that X is a P-space (resp., sequentially separable) if and only if so is F(X).
2
Content available remote On some forms of quasi-uniform convergence of transfinite sequence of multifunctions
Przemski M.
Commentationes Mathematicae
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2010
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Vol. 50, [Z] 1
3-21
EN
In this paper we introduce various forms of convergence of transfinite sequences of multifunctions with values in a quasi-uniform space. We also study some weak types of continuity for such multifunctions. Any such sequence of multifunctions generates the sequence of the sets of weak types of continuity points and the sequence of various types of cluster sets of members of such sequence. We study the connection between convergence of a transfinite sequences of multifunctions and convergence of the corresponding sequences of the sets of the weak continuity points and the sequences of cluster sets. Some of the presented results concern of general nets of multifunctions.
3
Content available remote Correcting theorem 1 from "When lines go bad in hyperspace"
Schlicker S., Bay C., Lembcke A.
Demonstratio Mathematica
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2009
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Vol. 42, nr 2
237-240
EN
This is in regards to the paper "When Lines go bad in hyperspace" by Christopher Bay, Amber Lembcke, and Steven Schlicker which appears in Demonstratio Mathematica, No. 3, Volume 38 (2005), p. 689-701. It has recently been brought to our attention that Theorem 1 from this paper is not correct. Please note that the main conclusions of the paper do not depend at all on this theorem. However, as the authors we feel it is our responsibility to bring this erroneous theorem to your attention. As stated in the paper, Theorem 1 intends to demonstrate that there can be infinitely many elements at a given location between two sets A and B.
4
Content available remote The spaces of closed convex sets in Euclidean spaces with the fell topology
Sakai K., Yang Z.
Bulletin of the Polish Academy of Sciences. Mathematics
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2007
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Vol. 55, no 2
139-143
EN
Let ConvF(Rn) be the space of all non-empty closed convex sets in Euclidean space Rn endowed with the Fell topology. We prove that ConvF(lRn) ≈ Rn x Q for every n > 1 whereas ConvF(R) ≈ R x I.
5
Content available remote On the Lifshits constant for hyperspaces
Leśniak K.
Bulletin of the Polish Academy of Sciences. Mathematics
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2007
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Vol. 55, no 2
155-160
EN
The Lifshits theorem states that any k-uniformly Lipschitz map with a bounded orbit on a complete metric space X has a fixed point provided k < x(X] where x(X) is the so-called Lifshits constant of X. For many spaces we have x(X) > 1. It is interesting whether we can use the Lifshits theorem in the theory of iterated function systems. Therefore we investigate the value of the Lifshits constant for several classes of hyperspaces.
6
Content available remote When lines go bad in hyperspace
Bay C., Lembcke A., Schlicker S.
Demonstratio Mathematica
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2005
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Vol. 38, nr 3
689--701
EN
Let H(Rn) denote the hyperspace of all non-empty compact subsets of Rn. The Hausdorff metric h provides a way to measure distances between two elements of H(Rn) and generates the complete metric space (H(Rn),h)). In this paper, we examine geometric properties of lines in H(Rn), as determined by the Hausdorff metric, and compare and contrast the properties of these lines with Euclidean lines in Rn. Several surprising properties of these objects will be highlighted.
7
Content available Hausdorff convergence of domains and their boundaries for shape optimal design
Holzleitner L.
Control and Cybernetics
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2001
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Vol. 30, no 1
23-44
EN
A question, which arises frequently in shape optimal design, is the convergence of domains. If the objective function is defined by using the solution of a PDE with boundary conditions, then also the convergence of the boundary is of importance. In this paper a criterion for a set of domains is defined, such that from [Omega_n] --> Omega follows [Gamma_n] --> Gamma if one is restricting to this set of domains. Moreover it is proved that this criterion is sharp, meaning that if [Omega_n] --> Omega implies [Gamma_n] --> Gamma holds for any sequence of this set, then this criterion has to be fulfilled. A similar criterion for the convergence of the Lebesgue measure of the boundaries my(Gamma_n) --> my(Gamma) is given.
8
Content available remote On non-injectivity of Borel functions selecting points from compact sets. 2
Balcerzak M., Peredko J., Pol R.
Bulletin of the Polish Academy of Sciences. Mathematics
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2001
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Vol. 49, no 2
103-105
EN
The note is a supplement to (1]. We refine a result from [1] on the non-injectivity of Borel selections on the hyperspaces and we discuss the relations of the results in [1] with some results obtained by Lecomte (4].
9
Uniform properties and hyperspaces of metrizable spaces
Costantini C., Vitolo P.
Journal of Applied Analysis
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1999
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Vol. 5, nr 2
187-196
EN
Two equivalent metrics can be compared, with respect to their uniform properties, in several different ways. We present some of them, and then use one of these conditions to characterize which metrics on a space induce the same lower Hausdorff topology on the hyperspace. Finally, we focus our attention to complete metrics.
10
Content available remote Inducible mappings between hyperspaces
Charatonik Janusz J., Charatonik W.
Bulletin of the Polish Academy of Sciences. Mathematics
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1998
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Vol. 46, no 1
5-9
EN
Given a continuum X we denote by [2^x] and C(X) the hyperspace of all nonempty compact subsets and of all nonempty subcontinua of X. For any two continua X and Y and a mapping [f : X --> Y let 2^f] and C(f) stand for the induced mappings between corresponding hyperspaces. A mapping g between the hypespaces is inducible is there exists a mapping f such that [g = 2^f] or g = C(f), respectively. Necessary and sufficient conditions are shown under which a given mapping g is inducible.
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