Some Properties of Hyperspaces with Applications to Continua Theory

A proper subcontinuum H of a continuum X is said to be a W-set provided for each continuous surjective function f from a continuum Y onto X, there exists a subcontinuum C of Y that maps entirely onto H. Hereditarily weakly confluent (HWC) mappings are those with the property that each restriction to a subcontinuum of the domain is weakly confluent. In this paper, we show that the monotone image of a W-set is a W-set and that there exists a continuum which is not in class W but which is the HWC image of a class W continuum.

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“…Without loss of generality, assume that f is surjective. Given any > 0 so that < 1 4 diam(f (H)), there is a δ > 0 so that if…”

Section: Monotone Maps and W -Setsmentioning

confidence: 99%

“…In what follows, a continuum is a compact, connected metric space, and the term map is used to denote a continuous function. It is known that monotone images of class W continua are in class W , as shown in [1]. In the summer of 2000, two questions arose related to this result.…”

Section: Introductionmentioning

confidence: 97%

Monotone images of W-sets and hereditarily weakly confluent images of continua

Hatch

Stanojević

2003

Publ Inst Math (Belgr)

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show abstract

“…Note that t 1 < s 2 . Proceeding as in the paragraph above it is possible to find a number t 2 ∈ (t 1 , s 2 ) and a point a 2 …”

Section: Lemma Let X Be a Nondegenerate Continuum And Let ε > 0 Be mentioning

confidence: 99%

Openness of induced projections

Charatonik¹,

Charatonik

Illanes³

2000

Proc. Amer. Math. Soc.

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Abstract. For continua X and Y it is shown that if the projection f : X × Y → X has its induced mapping C(f ) open, then X is C * -smooth. As a corollary, a characterization of dendrites in these terms is obtained.All spaces considered in this paper are assumed to be metric. A mapping means a continuous function. To exclude some trivial statements we assume that all considered mappings are not constant. A continuum means a compact connected space. Given a continuum X with a metric d, we let 2 X denote the hyperspace of all nonempty closed subsets of X equipped with the Hausdorff metric H defined by

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“…Theorem [4,Theorem 2.2]. // A is a continuum which has the covering property, then X is absolutely C*-smooth.…”

mentioning

confidence: 99%

“…Recently, it was proved that circle-like continua with no local separating subcontinua [4], and tree-like atriodic continua [5] have the covering property, and hence, they are in Class (W). In [5, Theorem 3.1 and Corollary 3.3], a very geometric method was introduced in order to check whether certain classes of continua have the covering property.…”

mentioning

confidence: 99%