On the surjective span and semispan of connected metric spaces

Lelek, A.

doi:10.4064/cm-37-1-35-45

Cited by 26 publications

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“…Variations of the span have been defined since then (cf [4,5,2]). In general it is difficult to evaluate the spans of a geometric object.…”

Section: Introductionmentioning

confidence: 99%

The spans of five star-like simple closed curves

Thelma¹

2007

Glas. Mat. Ser. III

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Abstract. Let X be a continuum, that is a compact, connected, nonempty metric space. The span of X is the least upper bound of the set of real numbers r which satisfy the following conditions: there exists a continuum, C, contained in X × X such that d(x, y) is larger than or equal to r for all (x, y) in C and p 1 (C) = p 2 (C), where p 1 , p 2 are the usual projection maps. The following question has been asked. If X and Y are two simple closed curves in the plane and Y is contained in the bounded component of the plane minus X, then is the span of X larger than the span of Y ? We define a set of simple closed curves, which we refer to as being five star-like. We answer this question in the affirmative when X is one of these simple closed curves. We calculate the spans of the simple closed curves in this collection and consider the spans of various geometric objects related to these simple closed curves.

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“…Variations of the span have been defined since then (cf [4,5,2]). In general it is difficult to evaluate the spans of a geometric object.…”

Section: Introductionmentioning

confidence: 99%

The spans of five star-like simple closed curves

Thelma¹

2007

Glas. Mat. Ser. III

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“…If (X, d) is a connected metric space, then define the surjective semispan of X, σ * 0 (X) (see [6]), to be…”

Section: Proof Observe Thatmentioning

confidence: 99%

“…In this paper, we demonstrate that no two of these versions of span agree for all simple triods or for all simple closed curves. We also include an example which violates a conjectured bound between two versions of span.A natural way to construct examples of metric spaces is to look at subsets of R 3 with the Euclidean metric (see, for instance, [5], [6], and Section 7 of this paper). In Section 3 we develop an alternative approach which allows one to construct a metric for a space with certain distances predetermined.…”

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confidence: 99%

“…A natural way to construct examples of metric spaces is to look at subsets of R 3 with the Euclidean metric (see, for instance, [5], [6], and Section 7 of this paper). In Section 3 we develop an alternative approach which allows one to construct a metric for a space with certain distances predetermined.…”

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confidence: 99%

“…Since then, several variants of his definition have been given. The most prevalent of these are the semispan (see [6]), the symmetric span (see [3]), and, for simple closed curves, the essential span (see [1]). …”

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confidence: 99%

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Equivalent metrics and the spans of graphs

Hoehn

Karassev

2009

Colloq. Math.

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Abstract. We present a result which affords the existence of equivalent metrics on a space having distances between certain pairs of points predetermined, with some restrictions. This result is then applied to obtain metric spaces which have interesting properties pertaining to the span, semispan, and symmetric span of metric continua. In particular, we show that no two of these variants of span agree for all simple closed curves or for all simple triods.1. Introduction. The span of a continuum was defined by A. Lelek in [4]. Since then, several variants of his definition have been given. The most prevalent of these are the semispan (see [6]), the symmetric span (see [3]), and, for simple closed curves, the essential span (see [1]).It has been asked (see, for instance, [1] and [2]) whether some of these different quantities always agree for certain classes of continua, particularly for simple triods and simple closed curves. In this paper, we demonstrate that no two of these versions of span agree for all simple triods or for all simple closed curves. We also include an example which violates a conjectured bound between two versions of span.A natural way to construct examples of metric spaces is to look at subsets of R 3 with the Euclidean metric (see, for instance, [5], [6], and Section 7 of this paper). In Section 3 we develop an alternative approach which allows one to construct a metric for a space with certain distances predetermined. Related results have been obtained in [7] and [8]. This enables us to prove in Section 6 the existence of spaces with interesting span properties without producing subsets of R 3 .

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The theory of continua

Odintsov¹,

Fedorchuk²

1994

J Math Sci

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On the surjective span and semispan of connected metric spaces

Cited by 26 publications

References 0 publications

The spans of five star-like simple closed curves

The spans of five star-like simple closed curves

Equivalent metrics and the spans of graphs

The theory of continua

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