Currently displaying 1 – 18 of 18

Showing per page

Order by Relevance | Title | Year of publication

Fans are not c-determined

Alejandro Illanes — 1999

Colloquium Mathematicae

A continuum is a compact connected metric space. For a continuum X, let C(X) denote the hyperspace of subcontinua of X. In this paper we construct two nonhomeomorphic fans (dendroids with only one ramification point) X and Y such that C(X) and C(Y) are homeomorphic. This answers a question by Sam B. Nadler, Jr.

Induced almost continuous functions on hyperspaces

Alejandro Illanes — 2006

Colloquium Mathematicae

For a metric continuum X, let C(X) (resp., 2 X ) be the hyperspace of subcontinua (resp., nonempty closed subsets) of X. Let f: X → Y be an almost continuous function. Let C(f): C(X) → C(Y) and 2 f : 2 X 2 Y be the induced functions given by C ( f ) ( A ) = c l Y ( f ( A ) ) and 2 f ( A ) = c l Y ( f ( A ) ) . In this paper, we prove that: • If 2 f is almost continuous, then f is continuous. • If C(f) is almost continuous and X is locally connected, then f is continuous. • If X is not locally connected, then there exists an almost continuous function f: X → [0,1] such that...

Pseudo-homotopies of the pseudo-arc

Alejandro Illanes — 2012

Commentationes Mathematicae Universitatis Carolinae

Let X be a continuum. Two maps g , h : X X are said to be pseudo-homotopic provided that there exist a continuum C , points s , t C and a continuous function H : X × C X such that for each x X , H ( x , s ) = g ( x ) and H ( x , t ) = h ( x ) . In this paper we prove that if P is the pseudo-arc, g is one-to-one and h is pseudo-homotopic to g , then g = h . This theorem generalizes previous results by W. Lewis and M. Sobolewski.

A continuum X such that C ( X ) is not continuously homogeneous

Alejandro Illanes — 2016

Commentationes Mathematicae Universitatis Carolinae

A metric continuum X is said to be continuously homogeneous provided that for every two points p , q X there exists a continuous surjective function f : X X such that f ( p ) = q . Answering a question by W.J. Charatonik and Z. Garncarek, in this paper we show a continuum X such that the hyperspace of subcontinua of X , C ( X ) , is not continuously homogeneous.

Whitney blocks in the hyperspace of a finite graph

Alejandro Illanes — 1995

Commentationes Mathematicae Universitatis Carolinae

Let X be a finite graph. Let C ( X ) be the hyperspace of all nonempty subcontinua of X and let μ : C ( X ) be a Whitney map. We prove that there exist numbers 0 < T 0 < T 1 < T 2 < < T M = μ ( X ) such that if T ( T i - 1 , T i ) , then the Whitney block μ - 1 ( T i - 1 , T i ) is homeomorphic to the product μ - 1 ( T ) × ( T i - 1 , T i ) . We also show that there exists only a finite number of topologically different Whitney levels for C ( X ) .

Homotopy properties of curves

Janusz Jerzy CharatonikAlejandro Illanes — 1998

Commentationes Mathematicae Universitatis Carolinae

Conditions are investigated that imply noncontractibility of curves. In particular, a plane noncontractible dendroid is constructed which contains no homotopically fixed subset. A new concept of a homotopically steady subset of a space is introduced and its connections with other related concepts are studied.

Factorwise rigidity of embeddings of products of pseudo-arcs

Mauricio E. Chacón-TiradoAlejandro IllanesRocío Leonel — 2012

Colloquium Mathematicae

An embedding from a Cartesian product of two spaces into the Cartesian product of two spaces is said to be factorwise rigid provided that it is the product of embeddings on the individual factors composed with a permutation of the coordinates. We prove that each embedding of a product of two pseudo-arcs into itself is factorwise rigid. As a consequence, if X and Y are metric continua with the property that each of their nondegenerate proper subcontinua is homeomorphic to the pseudo-arc, then X ×...

Continua with unique symmetric product

José G. AnayaEnrique Castañeda-AlvaradoAlejandro Illanes — 2013

Commentationes Mathematicae Universitatis Carolinae

Let X be a metric continuum. Let F n ( X ) denote the hyperspace of nonempty subsets of X with at most n elements. We say that the continuum X has unique hyperspace F n ( X ) provided that the following implication holds: if Y is a continuum and F n ( X ) is homeomorphic to F n ( Y ) , then X is homeomorphic to Y . In this paper we prove the following results: (1) if X is an indecomposable continuum such that each nondegenerate proper subcontinuum of X is an arc, then X has unique hyperspace F 2 ( X ) , and (2) let X be an arcwise connected...

Page 1

Download Results (CSV)