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Hyperspaces of two-dimensional continua

100%

Levin M. B. , Sternfeld Y.

Fundamenta Mathematicae

1996

tom 150

nr 1

17-24

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) = ∞.

Some metric properties of hyperspaces

100%

Bogdewicz A.

Demonstratio Mathematica

2000

tom Vol. 33, nr 1

135-149

Fans are not c-determined

100%

Illanes A.

Colloquium Mathematicum

1999

tom 81

nr 2

299-308

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.

Whitney Preserving Maps onto Dendrites

100%

Matsuhashi E.

Bulletin of the Polish Academy of Sciences. Mathematics

2012

tom Vol. 60, no 2

155--163

We prove the following results. (i) Let X be a continuum such that X contains a dense arc component and let D be a dendrite with a closed set of branch points. If f:X→D is a Whitney preserving map, then f is a homeomorphism. (ii) For each dendrite D′ with a dense set of branch points there exist a continuum X′ containing a dense arc component and a Whitney preserving map f′:X′→D′ such that f′ is not a homeomorphism.

On some forms of quasi-uniform convergence of transfinite sequence of multifunctions

75%

Przemski M.

Commentationes Mathematicae

2010

tom 50

nr 1

In this paper we introduce various forms of convergence of transfinite sequences of multifunctions with values in a quasi-uniform space. We also study some weak types of continuity for such multifunctions. Any such sequence of multifunctions generates the sequence of the sets of weak types of continuity points and the sequence of various types of cluster sets of members of such sequence. We study the connection between convergence of a transfinite sequences of multifunctions and convergence of the corresponding sequences of the sets of the weak continuity points and the sequences of cluster sets. Some of the presented results concern of general nets of multifunctions.