Whitney Map Research Articles

Hollowing to the hyperspace of subcontinua of dendrites

Let C(X) be the hyperspace of all subcontinua of a metric continuum X. An element A∈C(X)hollows toC(X) provided μ−1(μ(A))−{A} is connected and non-unicoherent for each Whitney map μ for C(X). In this paper the hollowing property of an element in hyperspaces is introduced as a refinement of the unicoherence and the connectedness and we characterize the elements A∈C(X) satisfying that A hollows to C(X) when X is a dendrite.

Topological properties in Whitney blocks

Let C(X) be the hyperspace of subcontinua of a continuum X. A Whitney block is a set of the form μ−1([s,t]), where μ:C(X)→[0,1] is a Whitney map and 0≤s<t≤1. In this paper, we study the following implication: if X has property P, then each Whitney block in C(X) has property P. We consider the following properties: connectedness im kleinen, being absolute neighborhood retract, local contractibility, and m-mutual aposyndesis.

Whitney blocks

Let C(X) be the hyperspace of subcontinua of a continuum X. A Whitney block in C(X) is a set of the form μ−1([s,t]), where μ:C(X)→[0,1] is a Whitney map and 0≤s<t<1. In this paper we study topological properties P satisfying the implication: if X has property P, then each Whitney block in C(X) has property P. We also consider the converse implication.We study properties related to local connectedness, arcwise connectedness, property of Kelley, contractibility, aposyndesis, unicoherence, etc.

Extending generalized Whitney maps

For metrizable continua, there exists the well-known notion of a Whitney map. If $X$ is a nonempty, compact, and metric space, then any Whitney map for any closed subset of $2^{X}$ can be extended to a Whitney map for $2^{X}$ [3, 16.10 Theorem]. The main purpose of this paper is to prove some generalizations of this theorem.

Contractibility is not a strong Whitney-reversible property for locally connected continua

We show an example of a noncontractible locally connected continuum X that admits a Whitney map μ for C(X) such that μ−1(t) is an absolute retract for each t∈(0,1).

The structure of Whitney blocks

Let X be a metric continuum and C(X) the hyperspace of subcontinua of X. A Whitney block is a set of the form μ−1([s,t]), where μ:C(X)→[0,1] is a Whitney map and 0≤s<t<1. In this paper we study continua for which Whitney blocks are homeomorphic to X×[0,1]. We characterize an arc, a simple closed curve and simple n-ods in terms of Whitney blocks. We also show that if X is arc-like, then each Whitney block for C(X) is 2-cell-like; and if X is circle-like, then each Whitney block of the form μ−1([0,t]) is ring-like.

On non-metric continua that support Whitney maps

We discuss the construction of non-metric continua that support Whitney maps and their properties. We indicate a technique for producing non-metric indecomposable continua that support Whitney maps from certain compact totally disconnected spaces each of which allows a self-homeomorphism all of whose orbits are dense. These are non-metric examples that have the property that each proper subcontinuum is metric. Both perfectly normal and non-perfectly normal examples are constructed. We describe techniques for producing large collections of non-homeomorphic continua that support Whitney maps. An example of a continuum every non-degenerate subcontinuum of which is non-metric that supports a Whitney map is constructed; an example of a continuum that does not support a Whitney map which is the union of two subcontinua each of which supports a Whitney map is constructed.

Elusive Examples of Non-Metrizable Continua which Admit a Whitney Map

Abstract The main purpose of this paper is to prove that the class of near locally connected continua contains no non-metrizable continuum 𝑋 which admits a Whitney map for 𝐶(𝑋). In particular, each near locally connected continuum 𝑋 which admits a Whitney map for 𝐶(𝑋) is metrizable.

D-continuum X admits a Whitney map for C(X) if and only if it is metrizable

The main purpose of this paper is to prove: a) a D- continuum X admits a Whitney map for C(X) if and only if it is metrizable, b) a continuum X admits a Whitney map for C 2 (X) if and only if it is metrizable.

Arcwise Connected Continua and Whitney Maps

Abstract Let 𝑋 be a non-metric continuum, and 𝐶(𝑋) be the hyperspace of subcontinua of 𝑋. It is known that there is no Whitney map on the hyperspace 2𝑋 for non-metric Hausdorff compact spaces 𝑋. On the other hand, there exist non-metric continua which admit and ones which do not admit a Whitney map for 𝐶(𝑋). In particular, a locally connected or a rimmetrizable continuum 𝑋 admits a Whitney map for 𝐶(𝑋) if and only if it is metrizable. In this paper we investigate the properties of continua 𝑋 which admit a Whitney map for 𝐶(𝑋) or for 𝐶2(𝑋).

Whitney maps-a non-metric case

It is shown that there is no Whitney map on the hyperspace $2^X$ for non-metrizable Hausdorff compact spaces X. Examples are presented of non-metrizable continua X which admit and ones which do not admit a Whitney map for C(X).

Multicoherence of Whitney levels

Let X be a Peano continuum and μ : C( X) → [0, 1] a Whitney map with C( X) the hyperspace of subcontinua of X. For a connected space Y, let r( Y) denote the multicoherence degree of Y. In this paper we prove: 1. (A) If s ⩽ t, then r( μ −1( s))⩾ r( μ −1( t)), 2. (B) r( μ −1( t)) is finite for every t > 0, 3. (C) if 0 < m ⩽ r( X), then there exists a Whitney map ν : C( X) → [0,1] and there exists t ϵ [0,1] such that r( ν −1( t)) = m, and 4. (D) X is a simple closed curve if and only if r( μ −1( t)) > 0 for every t < 1.

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.

Whitney maps for spaces of embedding hypersurfaces

The existence of Whitney maps is proved, and it is also shown that if X is a metrizable continuum, the Whitney map will be a trivial fibering with its own Hilbert cube.

Shore points in dendroids and conical pointed hyperspaces

If X is a continuum and μ a Whitney map for C( X), a subcontinuum Y of C( X) is μ-conical pointed if for some λ ∈[0, 1), the cone K(μ −1(λ) ⌢ Y) of μ −1(λ) ⌢ Y is homeomorphic with μ −1[λ, 1] ⌢ Y. This property generalizes the Roger's cone = hyperspace property. If X is a (smooth) dendroid, x ∈ X is a shore point if there is a sequence of subdendroids of X not containing x which converges to X. In this paper we give necessary and sufficient conditions on X, involving shore points, for C p ( X) to be μ-canonical pointed.

Simplicial Differential Forms with Poles

Shadow forms. Consider a simplicial complex K. Whitney associated to each simplicial cochain 4 a differential form v(4) on K. Whitney's map v from simplicial cochains to differential forms is a map of cochain complexes which induces the deRham isomorphism. We call the forms v(4) Whitney forms. They have been useful in several contexts, most notably integral geometry and rational homotopy theory. In this paper, we introduce a generalization of Whitney forms called shadow forms. We associate to certain simplicial chains g with respect to the first derived subdivision K' of K a shadow form w(() on K. Let us illustrate the construction of shadow forms when K is a single n-simplex, which we now denote by A\. In this case, w(g) is defined for k-simplices g of AX' which do not lie in the boundary of (or for linear combinations of these). For example, if A is the 2-simplex, then either of the 1-simplices (, or (2 pictured below has an associated shadow form.

A Note on Continuous Mappings and the Property of J. L. Kelley

In this paper, it is proved that if $X$ is a continuum and $\omega$ is any Whitney map for $C(X)$, then the following are equivalent: (1) $X$ has property [K]. (2) There exists a (continuous) mapping $F:X \times I \times [0,\omega (X)] \to C(X)$ such that $F(\{ x\} \times I \times \{ t\} ) = \{ A \in {\omega ^{ - 1}}(t)|x \in A\}$ for each $x \in X$ and $t \in [0,\omega (X)]$, where $I = [0,1]$. (3) For each $t \in [0,\omega (X)]$, there is an onto map $f:X \times I \to {\omega ^{ - 1}}(t)$ such that $f(\{ x\} \times I) = \{ A \in {\omega ^{ - 1}}(t)|x \in A\}$ for each $x \in X$. Some corollaries are obtained also.

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.