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Continuous pseudo-hairy spaces and continuous pseudo-fans

Janusz R. Prajs — 2002

Fundamenta Mathematicae

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 : X ̃ o n t o 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 = Y X , any subcontinuum of Y X intersecting two different elements of ℱ contains c, and ℱ is homeomorphic to X (with...

A fixed-point anomaly in the plane

Charles L. HagopianJanusz R. Prajs — 2005

Fundamenta Mathematicae

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.

On confluently graph-like compacta

Lex G. OversteegenJanusz R. Prajs — 2003

Fundamenta Mathematicae

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...

AANR spaces and absolute retracts for tree-like continua

Janusz Jerzy CharatonikJanusz R. Prajs — 2005

Czechoslovak Mathematical Journal

Continua that are approximative absolute neighborhood retracts (AANR’s) are characterized as absolute terminal retracts, i.e., retracts of continua in which they are embedded as terminal subcontinua. This implies that any AANR continuum has a dense arc component, and that any ANR continuum is an absolute terminal retract. It is proved that each absolute retract for any of the classes of: tree-like continua, λ -dendroids, dendroids, arc-like continua and arc-like λ -dendroids is an approximative absolute...

Arc property of Kelley and absolute retracts for hereditarily unicoherent continua

Janusz J. CharatonikWłodzimierz J. CharatonikJanusz R. Prajs — 2003

Colloquium Mathematicae

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.

Hyperspace retractions for curves

AbstractWe study retractions from the hyperspace of all nonempty closed subsets of a given continuum onto the continuum (which is naturally embedded in the hyperspace). Some necessary and some sufficient conditions for the existence of such a retraction are found if the continuum is a curve. It is shown that the existence of such a retraction for a curve implies that the curve is a uniformly arcwise connected dendroid, and that a universal smooth dendroid admits such a retraction. The existence...

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