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1
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Induced almost continuous functions on hyperspaces

100%
Illanes A.
Colloquium Mathematicum
|
2006
|
tom 105
|
nr 1
69-76
EN
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 $C(f)(A) = cl_{Y}(f(A))$ and $2^{f}(A) = cl_{Y}(f(A))$. 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 C(f) is almost continuous and f is not continuous.
2
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A characterization of dendroids by the n-connectedness of the Whitney levels

100%
Illanes A.
Fundamenta Mathematicae
|
1991-1992
|
tom 140
|
nr 2
157-174
EN
Let X be a continuum. Let C(X) denote the hyperspace of all subcontinua of X. In this paper we prove that the following assertions are equivalent: (a) X is a dendroid, (b) each positive Whitney level in C(X) is 2-connected, and (c) each positive Whitney level in C(X) is ∞-connected (n-connected for each n ≥ 0).
3
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The space of Whitney levels is homeomorphic to $l_2$

100%
Illanes A.
Colloquium Mathematicum
|
1993
|
tom 65
|
nr 1
1-11
4
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Fans are not c-determined

100%
Illanes A.
Colloquium Mathematicum
|
1999
|
tom 81
|
nr 2
299-308
EN
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.
5
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The openness of induced maps on hyperspaces

100%
Illanes A.
Colloquium Mathematicum
|
1997
|
tom 74
|
nr 2
219-224
6
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Factorwise rigidity of embeddings of products of pseudo-arcs

51%
Chacón-Tirado M., Illanes A., Leonel R.
Colloquium Mathematicum
|
2012
|
tom 128
|
nr 1
7-14
EN
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 × Y is factorwise rigid. This extends results of D. P. Bellamy and J. M. Lysko (for the case that X and Y are pseudo-arcs) and of K. B. Gammon (for the case that X is a pseudo-arc and Y is either a pseudo-circle or a pseudo-solenoid).
7
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Induced open projections and C*-smoothness

51%
Charatonik W., Illanes A., Martínez-de-la-Vega V.
Colloquium Mathematicum
|
2013
|
tom 132
|
nr 1
73-94
EN
We show that there exists a C*-smooth continuum X such that for every continuum Y the induced map C(f) is not open, where f: X × Y → X is the projection. This answers a question of Charatonik (2000).
8
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Cells and cubes in hyperspaces

32%
Illanes A.
Fundamenta Mathematicae
|
1988
|
tom 130
|
nr 1
57-65
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