Equivalent metrics and the spans of graphs

Abstract

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 .