On Unicoherency About a Simple Closed Curve

Abstract

1. The subject matter of this article is a discussion of a property of point-sets which is closely related to the properties of being connected and of being a unicoherent continuum, and is in a sense a generalization of both these properties. A unicoherent continuum is defined by Kuratowski t as one which cannot be expressed as the union of two continua whose divisor is not connected, and this definition is in common usage. The analogy of the definition to that of a continuum is apparent. Ilowever, a set that is not connected may be connected between some pair of its points. We therefore propose a corresponding definition of unicoherency. A set 11 in a imetric space is unicoherent about the simple closed curve J if, for every decomposition of J into closed arcs h and k by points a and b and for every decomposition of M into relatively closed sets H and K such that h C iH, k C K., and h K = IH = a + b, ther-e is a component of H K containin.g both a and b. (If M is closed or is the space itself, the word relatively is to be omitted.) It is this extension of the idea of unicoherent continuum that is to be discussed. That this may be regarded also as a natural extension of the definition of connectivity between two points is apparent when we recall that in a one-dimensional Euclidean space an interval is a sphere and two points form its frontier or the surface of the sphere. t The property of a set being unicoherent about a simple closed curve is also useful in formulating an intrinsic definition of a two-dimensional simplex, as will be seen later. Readers of II. Whitney's recent article on this subject ? will note the marked resemblance to the property of a simple closed curve being homologous to zero in a closed set, which is the basis of his article. In fact, the present article grew out of a search for purely point-set properties equivalent to Whitney's definition.