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Some examples of true $F_{σδ}$ sets

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EN
Let K(X) be the hyperspace of a compact metric space endowed with the Hausdorff metric. We give a general theorem showing that certain subsets of K(X) are true $F_{σδ}$ sets.
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Size levels for arcs

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We determine the size levels for any function on the hyperspace of an arc as follows. Assume Z is a continuum and consider the following three conditions: 1) Z is a planar AR; 2) cut points of Z have component number two; 3) any true cyclic element of Z contains at most two cut points of Z. Then any size level for an arc satisfies 1)-3) and conversely, if Z satisfies 1)-3), then Z is a diameter level for some arc.
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A Ramsey theorem for polyadic spaces

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A polyadic space is a Hausdorff continuous image of some power of the one-point compactification of a discrete space. We prove a Ramsey-like property for polyadic spaces which for Boolean spaces can be stated as follows: every uncountable clopen collection contains an uncountable subcollection which is either linked or disjoint. One corollary is that $(ακ)^ω$ is not a universal preimage for uniform Eberlein compact spaces of weight at most κ, thus answering a question of Y. Benyamini, M. Rudin and M. Wage. Another consequence is that the property of being polyadic is not a regular closed hereditary property.
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The ANR-property of hyperspaces with the Attouch-Wets topology

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We characterize metric spaces whose hyperspaces of non-empty closed, bounded, compact or finite subsets, endowed with the Attouch-Wets topology, are absolute (neighborhood) retracts.
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Whitney maps-a non-metric case

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It is shown that there is no Whitney map on the hyperspace $2^X$ for non-metrizable Hausdorff compact spaces X. Examples are presented of non-metrizable continua X which admit and ones which do not admit a Whitney map for C(X).
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The openness of induced maps on hyperspaces

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The space of ANR’s in $ℝ^n$

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The hyperspaces $ANR(ℝ^n)$ and $AR(ℝ^n)$ in $2^{ℝ^n} (n ≥ 3)$ consisting respectively of all compact absolute neighborhood retracts and all compact absolute retracts are studied. It is shown that both have the Borel type of absolute $G_{δσ δ}$-spaces and that, indeed, they are not $F_{σ δσ }$-spaces. The main result is that $ANR(ℝ^n)$ is an absorber for the class of all absolute $G_{δσ δ}$-spaces and is therefore homeomorphic to the standard model space $Ω_3$ of this class.
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The hyperspace of finite subsets of a stratifiable space

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It is shown that the hyperspace of non-empty finite subsets of a space X is an ANR (an AR) for stratifiable spaces if and only if X is a 2-hyper-locally-connected (and connected) stratifiable space.
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Hyperspaces of CW-complexes

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It is shown that the hyperspace of a connected CW-complex is an absolute retract for stratifiable spaces, where the hyperspace is the space of non-empty compact (connected) sets with the Vietoris topology.
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Hyperspaces of Peano continua of euclidean spaces

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If X is a space then L(X) denotes the subspace of C(X) consisting of all Peano (sub)continua. We prove that for n ≥ 3 the space $L(ℝ^n)$ is homeomorphic to $B^∞$, where B denotes the pseudo-boundary of the Hilbert cube Q.
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Hyperspace retractions for curves

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EN
CONTENTS 1. Introduction...................................5 2. Preliminaries.................................7 3. Hyperspace retractions.................9 4. Applications to selections............15 5. Applications to means.................18 References.....................................32
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Hereditarily weakly confluent induced mappings are homeomorphisms

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For a given mapping f between continua we consider the induced mappings between the corresponding hyperspaces of closed subsets or of subcontinua. It is shown that if either of the two induced mappings is hereditarily weakly confluent (or hereditarily confluent, or hereditarily monotone, or atomic), then f is a homeomorphism, and consequently so are both the induced mappings. Similar results are obtained for mappings between cones over the domain and over the range continua.
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Probability measure functors preserving infinite-dimensional spaces

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CONTENTS Introduction.................................................................................5 1. Decomposable topologies.......................................................6 2. Locally convex topologies......................................................10 3. Semilattices. Strong decomposability.....................................13 4. Convex topologies.................................................................15 5. Topologies on linearly ordered sets.......................................18 6. Topologies on lattices............................................................20 7. The Scott topology.................................................................26 8. Uniqueness of decomposition................................................28 9. Hyperspace topologies..........................................................32 10. The Vietoris topology...........................................................35 11. The Hausdorff metric topology.............................................37 12. The proximal topology..........................................................39 13. The Kuratowski convergence..............................................40 14. Uniqueness of decomposition for hypertopologies..............44 References................................................................................47
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