Michael Levin

Certain finite dimensional maps and their application to hyperspaces

In [8] Y. Sternfeld and this author gave a positive answer to the following longstanding open problem: Is the hyperspace (=the space of all subcontinua endowed with the Hausdorff metric) of a 2-dimensional continuum infinite dimensional? This result was improved in [9] where it was shown that for every positive integer numbern a 2-dimensional continuum contains a 1-dimensional subcontiuum with hyperspace of dimension ≥n. And it was asked there: Does a 2-dimensional continuum contain a 1-dimensional subcontinuum with infinite dimensional hyperspace? In this note we answer this question in the positive.Our proof applies maps with the following properties. A real valued mapf on a compactumX is called a Bing map if every continuum that is contained in a fiber off is hereditarily indecomposable.f is called ann-dimensional Lelek map if the union of all non-trivial continua which are contained in the fibers off isn-dimensional. It is shown that for dimX=n+1 the maps which are both Bing andn-dimensional Lelek maps form a denseGσ-subset of the function spaceC(X, I)

The space of subcontinua of a 2-dimensional continuum is infinite dimensional

Let X be a metric continuum and let C(X) denote the space of subcontinua of X with the Hausdorff metric. We settle a longstanding problem showing that if dimX = 2 then dimC(X) = ∞. The special structure and properties of hereditarily indecomposable continua are applied in the proof.

DIMENSION AND SUPERPOSITION OF CONTINUOUS FUNCTIONS

AbstractWe give a relatively short proof of the following theorem of Sternfeld: LetX be a compact metric space with dimX ≧ 2, and letX ⊂Rm be an embedding such that everyf ∈C(X) can be represented as

Mappings which are stable with respect to the property dim ƒ(X) ⩾ k

f(x_1 ,x_2 ,...,x_m ) = \sum\limits_{i = 1}^m {g_i (x_i ),} (x_1 ,x_2 ,...,x_m ) \in X,g_i \in

Monotone basic embeddings of hereditarily indecomposable continua

Thenm ≧ 2 dimX + 1.

Finite dimensional polish spaces are extreme boundaries of convex bodies in Euclidean space

Solving a problem by Arkhangelskiĭ, we construct a linear continuous open surjection L : Cp(X) → CP(Y) for compacta X and Y with 0 < dim X < dim Y < ∞. An example of nonopen linear continuous surjection of the space Cp[0, 1] onto itself is given. Related topics and results are discussed.

Bing maps and finite-dimensional maps

Abstract A continuous mapping ƒ : X → Y is called k-stable if for every metric space E that contains Y there exists a neighborhood U of ƒ in C(X, E) such that dim g(X) ⩾ k for all g in U. The paper is devoted to the study of k-stable maps on compact metric spaces.

Hyperspaces of two-dimensional continua

Abstract The Chogoshvili Claim states that for each k -dimensional compactum X in R n , there exists an ( n − k )-plane P in R n such that X is not removable from P . This means that for some e > 0, every map f : X → R n with ∥ x − f ( x )∥ e for all x ϵ X , has the property that f ( X ) ∩ P ≠ φ . A counterexample to this claim has recently been constructed by A. Dranishnikov. Our results show, among other things, that each 2-dimensional LC 1 compactum, and hence each 2-dimensional disk, obeys the claim. To help indicate the sharpness of the preceding, we also provide a local path-connectification of Dranishnikovs example.