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1
Content available remote

Some examples of true $F_{σδ}$ sets

100%
Balcerzak M., Darji U. B.
Colloquium Mathematicum
|
2000
|
tom 86
|
nr 2
203-207
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.
2
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Size levels for arcs

80%
Sam B. Jr. Nadler, West T.
Fundamenta Mathematicae
|
1992
|
tom 141
|
nr 3
243-255
EN
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.
3
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A Ramsey theorem for polyadic spaces

80%
Bell M. G.
Fundamenta Mathematicae
|
1996
|
tom 150
|
nr 2
189-195
EN
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.
4
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The ANR-property of hyperspaces with the Attouch-Wets topology

80%
Voytsitskyy R.
Open Mathematics
|
2008
|
tom 6
|
nr 2
228-236
EN
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.
5
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Whitney maps-a non-metric case

80%
Charatonik J. J., Charatonik W. J.
Colloquium Mathematicum
|
2000
|
tom 83
|
nr 2
305-307
EN
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).
6
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The openness of induced maps on hyperspaces

80%
Illanes A.
Colloquium Mathematicum
|
1997
|
tom 74
|
nr 2
219-224
7
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The space of ANR’s in $ℝ^n$

71%
Dobrowolski T., Rubin L. R.
Fundamenta Mathematicae
|
1994-1995
|
tom 146
|
nr 1
31-58
EN
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.
8
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The hyperspace of finite subsets of a stratifiable space

71%
Cauty R., Guo B.-L., Sakai K.
Fundamenta Mathematicae
|
1995
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tom 147
|
nr 1
1-9
EN
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.
9
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Hyperspaces of CW-complexes

61%
Guo B.-L., Sakai K.
Fundamenta Mathematicae
|
1993
|
tom 143
|
nr 1
23-40
EN
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.
10
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Hyperspaces of Peano continua of euclidean spaces

51%
Gladdines H., van Mill J.
Fundamenta Mathematicae
|
1993
|
tom 142
|
nr 2
173-188
EN
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.
11
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Hyperspace retractions for curves

43%
Charatonik J. J., Charatonik W. J., Omiljanowski K., Prajs J. R.
Instytut Matematyczny Polskiej Akademii Nauk
EN
CONTENTS 1. Introduction...................................5 2. Preliminaries.................................7 3. Hyperspace retractions.................9 4. Applications to selections............15 5. Applications to means.................18 References.....................................32
12
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Hereditarily weakly confluent induced mappings are homeomorphisms

41%
Charatonik J. J., Charatonik W. J.
Colloquium Mathematicum
|
1998
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tom 75
|
nr 2
195-203
EN
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.
13
Content available remote

Probability measure functors preserving infinite-dimensional spaces

41%
Nhu N. T., Sakai K.
Colloquium Mathematicum
|
1996
|
tom 70
|
nr 2
291-304
14
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Decomposition of topologies on lattices and hyperspaces

38%
Costantini C., Vitolo P.
Instytut Matematyczny Polskiej Akademii Nauk
EN
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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