On the oscillation of a continuum at a point

Abstract

1. The concept of oscillatiLon of a continuum at a given point was introduced by S. Ilazurkiewicz in his paper on Jordan curvest and the idea has since been used frequently by other writerst. Less attention has been given to the general concept, however, than to the special case where the oscillation is zero, a property which H. Hahn has shown to be equivalent to the property of connectedness im kleinen. In this article several new properties of the oscillatory function are developed. In particular it is shown that the oscillation is an upper semicontinuous function and therefore has the well known characteristics of pointwise discontinuous functions. Furthermore, the behavior of the function in the neighborhood of points where its value is greater than zero is investigated. (See ?? 11-15.) The results are obtained by the introduction of an auxiliary function v(a) (? 3) and oscillatory sub-sets (?? 7-10) of a continuum, both of which have interesting properties in themselves. The work is confined to continua, although several of the theorems can be readily extended to non-closed connected sets. 2. Notation. The ordinary notation of the theory of aggregates is employed, with the following modifications. If A is the common part of a system of aggregates {C}, we write A Dvv[C]. If A is a real part of B, we write AC B. If A is a part of B and may be identical with B, we write A C B. 3. Definitions. Let A be any continuum and a.-A. Let VO (a) denote the set of points of A whose distance from a is less than s, d> 0. Let CO denote a subcontinuum of A which contains all points of V8 (a). The lower bound of the diameters of all such sets 0a, for all > 0, is denoted by TA(a), or simply a(a). For convenience the previous sentence may be written