Wyniki wyszukiwania

1

On finite-dimensional maps and other maps with "small" fibers

100%

Sternfeld Y.

Fundamenta Mathematicae

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1995

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tom 147

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nr 2

127-133

EN

We prove that if f is a k-dimensional map on a compact metrizable space X then there exists a σ-compact (k-1)-dimensional subset A of X such that f|X∖A is 1-dimensional. Equivalently, there exists a map g of X in $I^k$ such that dim(f × g)=1. These are extensions of theorems by Toruńczyk and Pasynkov obtained under the additional assumption that f(X) is finite-dimensional. These results are then extended to maps with fibers restricted to some classes of spaces other than the class of k-dimensional spaces. For example: if f has weakly infinite-dimensional fibers then dim(f|X∖A) ≤ 1 for some σ-compact weakly infinite-dimensional subset A of X. The proof applies essentially the properties of hereditarily indecomposable continua.

2

Hyperspaces of two-dimensional continua

63%

Levin M. B., Sternfeld Y.

Fundamenta Mathematicae

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1996

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tom 150

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nr 1

17-24

EN

Let X be a compact metric space and let C(X) denote the space of subcontinua of X with the Hausdorff metric. It is proved that every two-dimensional continuum X contains, for every n ≥ 1, a one-dimensional subcontinuum $T_n$ with $dim C (T_n) ≥ n$. This implies that X contains a compact one-dimensional subset T with dim C (T) = ∞.

3

The structure of atoms (hereditarily indecomposable continua)

51%

Ball R. N., Hagler J. N., Sternfeld Y.

Fundamenta Mathematicae

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1998

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tom 156

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nr 3

261-278

EN

Let X be an atom (= hereditarily indecomposable continuum). Define a metric ϱ on X by letting $ϱ(x,y) = W(A_{xy})$ where $A_{x,y}$ is the (unique) minimal subcontinuum of X which contains x and y and W is a Whitney map on the set of subcontinua of X with W(X) = 1. We prove that ϱ is an ultrametric and the topology of (X,ϱ) is stronger than the original topology of X. The ϱ-closed balls C(x,r) = {y ∈ X:ϱ ( x,y) ≤ r} coincide with the subcontinua of X. (C(x,r) is the unique subcontinuum of X which contains x and has Whitney value r.) It is proved that for any two (nontrivial) atoms and any Whitney maps on them, the corresponding ultrametric spaces are isometric. This implies in particular that the combinatorial structure of subcontinua is identical in all atoms. The set M(X) of all monotone upper semicontinuous decompositions of X is a lattice when ordered by refinement. It is proved that for two atoms X and Y, M(X) is lattice isomorphic to M(Y) if and only if X is homeomorphic to Y.

Ograniczanie wyników

3 Fundamenta Mathematicae

3 Sternfeld Y.

1 Ball R. N.

1 Hagler J. N.

1 Levin M. B.

1 1998

1 1996

1 1995

On finite-dimensional maps and other maps with "small" fibers

Hyperspaces of two-dimensional continua

The structure of atoms (hereditarily indecomposable continua)