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A note on inverse limits of continuous images of arcs.

Ivan Loncar (1999)

Publicacions Matemàtiques

The main purpose of this paper is to prove some theorems concerning inverse systems and limits of continuous images of arcs. In particular, we shall prove that if X = {Xa, pab, A} is an inverse system of continuous images of arcs with monotone bonding mappings such that cf (card (A)) ≠ w1, then X = lim X is a continuous image of an arc if and only if each proper subsystem {Xa, pab, B} of X with cf(card (B)) = w1 has the limit which is a continuous image of an arc (Theorem 18).

A note on singular homology groups of infinite products of compacta

Kazuhiro Kawamura (2002)

Fundamenta Mathematicae

Let n be an integer with n ≥ 2 and X i be an infinite collection of (n-1)-connected continua. We compare the homotopy groups of Σ ( ∏ i X i ) with those of ∏ i Σ X i (Σ denotes the unreduced suspension) via the Freudenthal Suspension Theorem. An application to homology groups of the countable product of the n(≥ 2)-sphere is given.

A note on the paper ``Smoothness and the property of Kelley''

Gerardo Acosta, Álgebra Aguilar-Martínez (2007)

Commentationes Mathematicae Universitatis Carolinae

Let X be a continuum. In Proposition 31 of J.J. Charatonik and W.J. Charatonik, Smoothness and the property of Kelley, Comment. Math. Univ. Carolin. 41 (2000), no. 1, 123–132, it is claimed that L ( X ) = ⋂ p ∈ X S ( p ) , where L ( X ) is the set of points at which X is locally connected and, for p ∈ X , a ∈ S ( p ) if and only if X is smooth at p with respect to a . In this paper we show that such equality is incorrect and that the correct equality is P ( X ) = ⋂ p ∈ X S ( p ) , where P ( X ) is the set of points at which X is connected im kleinen. We also use the correct...

A weakly chainable uniquely arcwise connected continuum without the fixed point property

Mirosław Sobolewski (2015)

Fundamenta Mathematicae

A continuum is a metric compact connected space. A continuum is chainable if it is an inverse limit of arcs. A continuum is weakly chainable if it is a continuous image of a chainable continuum. A space X is uniquely arcwise connected if any two points in X are the endpoints of a unique arc in X. D. P. Bellamy asked whether if X is a weakly chainable uniquely arcwise connected continuum then every mapping f: X → X has a fixed point. We give a counterexample.

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...

Absolute end points of irreducible continua

Janusz Jerzy Charatonik (1993)

Mathematica Bohemica

A concept of an absolute end point introduced and studied by Ira Rosenholtz for arc-like continua is extended in the paper to be applied arbitrary irreducible continua. Some interrelations are studied between end points, absolute end points and points at which a given irreducible continuum is smooth.

Absolute n-fold hyperspace suspensions

Sergio Macías, Sam B. Nadler, Jr. (2006)

Colloquium Mathematicae

The notion of an absolute n-fold hyperspace suspension is introduced. It is proved that these hyperspaces are unicoherent Peano continua and are dimensionally homogeneous. It is shown that the 2-sphere is the only finite-dimensional absolute 1-fold hyperspace suspension. Furthermore, it is shown that there are only two possible finite-dimensional absolute n-fold hyperspace suspensions for each n ≥ 3 and none when n = 2. Finally, it is shown that infinite-dimensional absolute n-fold hyperspace suspensions...

Absolutely terminal continua and confluent mappings

Janusz Jerzy Charatonik (1991)

Commentationes Mathematicae Universitatis Carolinae

Interrelations between three concepts of terminal continua and their behaviour, when the underlying continuum is confluently mapped, are studied.

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