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1
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Continuous pseudo-hairy spaces and continuous pseudo-fans

100%
Prajs J.
Fundamenta Mathematicae
|
2002
|
tom 171
|
nr 2
101-116
EN
A compact metric space X̃ is said to be a continuous pseudo-hairy space over a compact space X ⊂ X̃ provided there exists an open, monotone retraction $r: X̃ {onto \atop ⟶ } X$ such that all fibers $r^{-1}(x)$ are pseudo-arcs and any continuum in X̃ joining two different fibers of r intersects X. A continuum $Y_{X}$ is called a continuous pseudo-fan of a compactum X if there are a point $c ∈ Y_{X}$ and a family ℱ of pseudo-arcs such that $⋃ ℱ = Y_{X}$, any subcontinuum of $Y_{X}$ intersecting two different elements of ℱ contains c, and ℱ is homeomorphic to X (with respect to the Hausdorff metric). It is proved that for each compact metric space X there exist a continuous pseudo-hairy space over X and a continuous pseudo-fan of X.
2
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A fixed-point anomaly in the plane

63%
Hagopian C., Prajs J.
Fundamenta Mathematicae
|
2005
|
tom 186
|
nr 3
233-249
EN
We define an unusual continuum M with the fixed-point property in the plane ℝ². There is a disk D in ℝ² such that M ∩ D is an arc and M ∪ D does not have the fixed-point property. This example answers a question of R. H. Bing. The continuum M is a countable union of arcs.
3
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On confluently graph-like compacta

63%
Oversteegen L., Prajs J.
Fundamenta Mathematicae
|
2003
|
tom 178
|
nr 2
109-127
EN
For any class 𝒦 of compacta and any compactum X we say that: (a) X is confluently 𝒦-representable if X is homeomorphic to the inverse limit of an inverse sequence of members of 𝒦 with confluent bonding mappings, and (b) X is confluently 𝒦-like provided that X admits, for every ε >0, a confluent ε-mapping onto a member of 𝒦. The symbol 𝕃ℂ stands for the class of all locally connected compacta. It is proved in this paper that for each compactum X and each family 𝒦 of graphs, X is confluently 𝒦-representable if and only if X is confluently 𝒦-like. We also show that for any compactum the properties of: (1) being confluently graph-representable, and (2) being 1-dimensional and confluently 𝕃ℂ-like, are equivalent. Consequently, all locally connected curves are confluently graph-representable. We also conclude that all confluently arc-like continua are homeomorphic to inverse limits of arcs with open bonding mappings, and all confluently tree-like continua are absolute retracts for hereditarily unicoherent continua.
4
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Arc property of Kelley and absolute retracts for hereditarily unicoherent continua

51%
Charatonik J., Charatonik W., Prajs J.
Colloquium Mathematicum
|
2003
|
tom 97
|
nr 1
49-65
EN
We investigate absolute retracts for hereditarily unicoherent continua, and also the continua that have the arc property of Kelley (i.e., the continua that satisfy both the property of Kelley and the arc approximation property). Among other results we prove that each absolute retract for hereditarily unicoherent continua (for tree-like continua, for λ-dendroids, for dendroids) has the arc property of Kelley.
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