Concerning acyclic continuous curves

Abstract

In this paper, we propose to use the word acyclic in place of the phrase containing no simple closed curve. That is, an acyclic continuous curve is a continuous curve containing no simple closed curve. Acyclic continuous curves have been studied by Mazurkiewicz,T R. L. Wilder,? R. L. Moore,lI and the author.? As a result of Theorem 1, it follows that certain internal properties of an acyclic continuous curve which have been proved by the above authors for plane curves, are also possessed by curves in n-dimensional space. However, in the present paper, only plane point sets are considered, unless otherwise stated.