On the Construction of Simple Arcs

Abstract

1. In. this paper it is proposed, in the first place, to give a simple, complete, and essentially new demonstration for the well-known theorem that every compact locally connected continuum is arewise connected. All previous proofs for this fundamental proposition with which the author is familiar are alike in that the arc is constructed by means of a monotone decreasing sequence of simple chains of open sets or regions. Due to the fact that two successive links in such chains necessarily overlap in more than one point, thereby leaving open the possibility that the next chain in the sequence may oscillate widely within a given chain, a certain element of elusiveness seems unavoidable when these chains are used to construct arcs; and in order to completely eliminate all such possibilities of oscillationl and make the argument entirely convincing, some tedious details are necessary. In the proof to be given below these difficulties are avoided by making use, in the construction, of chains of continua, called regulai chainsr, defined as follows: C is a simple regular chain of continua joining two points a and b if C is the sum of a finite number of compact continua a C X1, X2, * * , Xn D b such that any two successive links have exactly one common point, links that are not successive have nothing in common, and only the first and last links contain a and b respectively. If 8 (Xi) 0. Thus by simple recursion we can set up a monotone decreasing sequence of these chains from a to b such that the norm, E, approaches 0, and their product is easily proved to be an arc from a to b. Clearly such regular chains represent better approximations to arcs than do the simple chains of open sets, because since each intersection point of two links in the chain separates the chain between a and b, every such point in any chain in the sequence must belong to the product set and thus to the fixnal arc. No difficulties of oscillation can arise in this construction because, there being only one common point for any two successive links of a given chain, any later