On the topology of curves II

Lelek, A.

doi:10.4064/fm-70-2-131-138

Cited by 43 publications

(14 citation statements)

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“…If each point of a continuum X has arbitrary small neighborhood with finite boundary, then X is said to be regular. X is suslinian provided any collection of mutually disjoint nondegenerate subcontinua of X is at most countable [6]. For a nondegenerate continuum we have the following implications: BaC n aU n (see [5], p. 260) .…”

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confidence: 99%

Remark on mappings not raising dimension of curves

Krasinkiewicz¹

1974

Pacific J. Math.

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confidence: 99%

Remark on mappings not raising dimension of curves

Krasinkiewicz¹

1974

Pacific J. Math.

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“…We have g(a 0 ) < 0 and g((ii) > 1, by (11), and thus r 2g(ao) = 0andr 2 g(ai) = 1. Ifx G p' 1 (y), then hrip(x) = hr x (y) = h{y) = c. Therefore/(£> _1 (y)) C \c] X I, by (12). Since the continuum p~l(y) contains both points a 0 and ai, its image under/ contains the end-points/(ao) = (c, 0) and/(ai) = (c, 1) of the arc {c} X /, by (12).…”

Section: Theorem a Compactant X Is Non-suslinian If And Only If Thermentioning

confidence: 92%

Weakly Confluent Mappings and Atriodic Suslinian Curves

Cook¹,

Lelek²

1978

Can. j. math.

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There are theorems in which some classes of topological spaces are characterized by means of properties of mappings of these spaces into a single space. For example, it is well known that a compactum X is at most n-dimensional if and only if no mapping of X irto an (n + l)-cube has a stable value [5, Theorems VI. 1-2, pp. 75-77]. Also, a curve X is tree-like if and only if no mapping of X into a figure eight is homotopically essential [1, Theorem 1, pp. 74-75; 8, p. 91]. By a curve we mean any at most 1-dimensional continuum; a continuum is a connected compactum; a compactum is a compact metric space, and a mapping is a continuous function. The aim of the present paper is to prove another theorem of this type. We distinguish a class of curves and show that it is characterized by imposing the condition that no weakly confluent mapping [13] can transform the given curve onto a simple triod (see 2.4). A related result is applied to a generalized branch-point covering theorem (see 3.2). In addition, two results are obtained in which we establish some characterizations of weakly confluent images and preimages of the product of the Cantor set and an arc (see 1.1 and 2.2). Continua that are such images turn out to be identical with regular curves (see 1.3).

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“…A continuum is a connected compact metrizable space with more than one point. A continuum is Suslinian if there is no uncountable collection of pairwise disjoint subcontinua [10]. Suslinian continua are frequently called curves because they are 1dimensional.…”

Section: Preliminariesmentioning

confidence: 99%