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AANR spaces and absolute retracts for tree-like continua

Janusz Jerzy Charatonik, Janusz 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...

Affine group acting on hyperspaces of compact convex subsets of ℝⁿ

Sergey A. Antonyan, Natalia Jonard-Pérez (2013)

Fundamenta Mathematicae

For every n ≥ 2, let cc(ℝⁿ) denote the hyperspace of all nonempty compact convex subsets of the Euclidean space ℝⁿ endowed with the Hausdorff metric topology. Let cb(ℝⁿ) be the subset of cc(ℝⁿ) consisting of all compact convex bodies. In this paper we discover several fundamental properties of the natural action of the affine group Aff(n) on cb(ℝⁿ). We prove that the space E(n) of all n-dimensional ellipsoids is an Aff(n)-equivariant retract of cb(ℝⁿ). This is applied to show that cb(ℝⁿ) is homeomorphic...

Arc property of Kelley and absolute retracts for hereditarily unicoherent continua

Janusz J. Charatonik, Włodzimierz J. Charatonik, Janusz 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.

Borsuk-Sieklucki theorem in cohomological dimension theory

Margareta Boege, Jerzy Dydak, Rolando Jiménez, Akira Koyama, Evgeny V. Shchepin (2002)

Fundamenta Mathematicae

The Borsuk-Sieklucki theorem says that for every uncountable family X α α ∈ A of n-dimensional closed subsets of an n-dimensional ANR-compactum, there exist α ≠ β such that d i m ( X α ∩ X β ) = n . In this paper we show a cohomological version of that theorem: Theorem. Suppose a compactum X is c l c ℤ n + 1 , where n ≥ 1, and G is an Abelian group. Let X α α ∈ J be an uncountable family of closed subsets of X. If d i m G X = d i m G X α = n for all α ∈ J, then d i m G ( X α ∩ X β ) = n for some α ≠ β. For G being a countable principal ideal domain the above result was proved by Choi and Kozlowski...

Classical-type characterizations of non-metrizable ANE(n)-spaces

Valentin Gutev, Vesko Valov (1994)

Fundamenta Mathematicae

The Kuratowski-Dugundji theorem that a metrizable space is an absolute (neighborhood) extensor in dimension n iff it is L C n - 1 C n - 1 (resp., L C n - 1 ) is extended to a class of non-metrizable absolute (neighborhood) extensors in dimension n. On this base, several facts concerning metrizable extensors are established for non-metrizable ones.

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