Whitney Levels Research Articles

Singular decomposable continua

In this paper, we first provide an argument for the method used in [7] and [10] to blow up a point inside a subarc of a one-dimensional continuum to an arbitrary continuum. Next, we give an example of s Wilder continuum containing no strongly Wilder continua, no continuum-wise Wilder continua, no semiaposyndetic continua and no D⁎-continua. Also, we provide an example of a continuum such that each positive Whitney level of the hyperspace of the continuum is strongly Wilder, although the continuum itself does not contain any Wilder continua.

More on Whitney levels of some decomposable continua

In this paper, we show that there exists a non-D-continuum such that each positive Whitney level of the hyperspace of subcontinua of the continuum is both D⁎ and Wilder. We show that the property of being continuum-wise Wilder is not a Whitney property, while it is a Whitney reversible property. Furthermore, we introduce the new class of continua: closed set-wise Wilder continua. This class is larger than the class of continuum chainable continua and smaller than the class of continuum-wise Wilder continua. In addition to the above results, we show that the Cartesian product of two closed set-wise Wilder continua is close set-wise Wilder.

Some characterizations of the internal structure of Whitney levels

Let X be a continuum, and let C(X) denote the hyperspace of all subcontinua of X. It is known that there exist monotone maps μ from C(X) into [0, ∞) such that μ({x}) = 0 for each x ∈ X, and if A is a proper subcontinuum of B, then μ(A) < μ(B). The subcontinua μ−1(t) of C(X) are called Whitney levels of C(X). In this paper, a class of closed subsets of X is employed to characterize the Whitney levels of C(X) possessing one of the following properties: irreducibility, decomposability, being a Wilder continuum, aposyndesis, semiaposyndesis, n-aposyndesis, finite aposyndesis, and connectedness colocal.

Pseudo-arcs in the hyperspace of subcontinua of the pseudo-arc

We construct a pseudo-arc Q in the hyperspace of subcontinua of the pseudo-arc such that Q does not contain singletons and it is not contained in any Whitney level. This answers a question by Norman S. Passmore.

Some decomposable continua and Whitney levels of their hyperspaces

We introduce the new class of continua; D⁎⁎-continua. The classes of Wilder continua and D⁎-continua are strictly contained in the class of D⁎⁎-continua. Also, the class of D-continua properly contains the class of D⁎⁎-continua. Using D⁎⁎-continua, we give the negative answer to [2, Question 2]. Furthermore, we prove that being a Wilder continuum, being a D-continuum, being a D⁎-continuum and being a D⁎⁎-continuum are Whitney properties.

Non-weak cut, shore and non-cut points in Whitney levels

We show that the property of having only non-weak cut points is a Whitney property, and that having only shore points is not a Whitney property. We also note that the property of having only non-cut (non-weak cut, shore) points is not a Whitney reversible property. This complements a result due to E. Matsuhashi.

Dendrites which are determined by their positive Whitney levels

In this paper we prove that the property of a continuum of being determined by the positive Whitney levels of its hyperspace of subcontinua characterizes the dendrites whose set of endpoints is closed in the class of dendrites. This answers a question by Alejandro Illanes and Rocío Leonel.

Nonblockers in homogeneous continua

Given a continuum X, an element A∈2X∖{X} is said to be a set that does not block singletons of X provided that for every point x∈X∖A, there exists an order arc α:[0,1]→C(X) such that α(0)={x}, α(1)=X and α(t)∩A=∅ for all t∈[0,1). If A does not block singletons of X, we say that A is a nonblocker of X. In this paper, we present some properties about the sets that do not block singletons and, for a homogeneous curve X, we use the set NB(F1(X)) to construct a continuous decomposition of X that is a homogeneous curve and a Whitney level of C(X), where its elements are homogeneous, acyclic, hereditarily unicoherent and indecomposable continuum. Also, we prove that if X is a continuum such that NB(F1(X))=F1(X), then the next three properties are equivalent: 1) X is a homogeneous continuum, 2) X has the property that for all x∈X, there exists a proper subcontinuum A of X such that x∈Int(A), and 3) X≈S1.

Whitney levels in hyperspaces of non-metrizable continua

Let X be a Hausdorff continuum (a nondegenerate, compact, connected, Hausdorff space). Let C(X) (respectively F1(X)) denote the hyperspace of its subcontinua (respectively, its one-point sets), endowed with the Vietoris topology. In this paper we introduce the definition of Whitney levels in C(X) and discuss some basic properties. With this definition, the subsets F1(X) and {X} of C(X) are Whitney levels in C(X), so we call them trivial Whitney levels. In the particular case when X is a generalized arc, we give a condition for the existence of non-trivial Whitney levels in its hyperspace of subcontinua. Finally, we apply this result to the study of Whitney levels in C(X) when X is the Long Arc and the Lexicographic Square.

Property of Kelley for the Cartesian products and hyperspaces

A continuum X X having the property of Kelley is constructed such that neither X × [ 0 , 1 ] X \times [0,1] , nor the hyperspace C ( X ) C(X) , nor small Whitney levels in C ( X ) C(X) have the property of Kelley. This answers several questions asked in the literature.

The cone [formula omitted] hyperspace property, a characterization

Let X be a continuum. Let C(X) be the hyperspace of subcontinua of X. We say that X is said to have the cone = hyperspace property if there exists a homeomorphism h :C(X)→ Cone(X) such that h(X)= vertex of ( Cone(X) ) and h({x})=(x,0) for each x∈X. In this paper we prove. Theorem. Let X be a finite-dimensional continuum. Then the following are equivalent : (a) X has the cone = hyperspace property, and (b) there is a selection s :C(X)−{X}→X such that, for every Whitney level A for C(X), s| A : A→X is a homeomorphism. We show some consequences of this theorem.

Uniform pathwise connectedness and Whitney levels

It is proved that the property of being a uniformly pathwise connected continuum is a Whitney property, but is not a Whitney-reversible property.

Determining finite graphs by their large Whitney levels

Let G be a finite connected graph and C(G) the hyperspace of all subcontinua of G. A Whitney map is a continuous function μ:C(G)→[0,1] such that μ({p})=0 for each p∈G,μ(G)=1 and A⊂B≠A implies that μ(A)<μ(B). A large Whitney level is a set of the form μ−1(t) where 1>t>max{μ(S): S is a proper nonempty connected subgraph of G}. In this paper, we prove the following: Theorem. Let G and H be finite connected graphs with no cut points, then G and H are isomorphic if and only if large Whitney levels for C(G) are homeomorphic to large Whitney levels for C(H).

Monotone and open Whitney maps defined in 2 X

Some equivalent conditions on a Whitney map for 2 X to be open (or monotone) are given. It is shown that the existence of a monotone Whitney map for 2 X implies that the space of Whitney levels in 2 X is pathwise connected. Three examples showing that the contractibility of a dendroid X is independent of the existence of open (or monotone) Whitney maps for 2 X are given and some questions concerning this subject are asked.

A selection theorem for topological convex structures

A continuous selection theorem has been obtained for multivalued functions, the values of which are convex sets of certain synthetic convex structures. Applications are given related with superextensions, (semi)lattices, spaces of order arcs, trees, Whitney levels in hyperspaces, and geometric topology. Applications to traditional convexity in vector spaces involve Beer’s approximation theorem and a fixed point theorem of Dugundji-Granas. Some other applications (a.o. an invariant arc theorem) appear elsewhere.

The space of Whitney levels is homeomorphic to $l_2$

The space of Whitney levels is homeomorphic to $l_2$

Continua as positive Whitney levels

It is shown that every continuum is a positive Whitney level of some continuum.

A characterization of dendroids by the n-connectedness of the Whitney levels

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 l

The space of Whitney levels

Let X be a continuum and let C( X) be the space of subcontinua of X. In this paper we consider the spaces W( X) = { u: C( X)→ R ∣ u is a Whitney map and u( X) = 1} with the “sup metric” and, N( X)={ u -1( t)∈ C( C( X)): u∈ W( X) and 0⩽ t⩽1}. We define a natural order for N( X) and we prove that if there is a homeomorphism from N( X) onto N( Y) which preserves order, then X is homeomorphic to Y. Also we prove that W( X) is always homeomorphic to l 2 (this answers a question by Nadler).

Whitney levels and certain order arc spaces

Let X be a metric continuum, μ: C(X)→[0, 1] a Whitney map with C(X) the hyperspace of subcontinua of X, and Λ l s ( X)={order arcs α∣ α(0)∈ μ −1([ s, t]) and α(1)={ X}}. In this paper, we study the relationships between the Whitney intervals μ −1([ s,t]) and the order arc spaces Λ l s ( X). We give applications to Whitney properties and obtain some shape and homotopy results about Whitney levels.