Almost Cyclic Elements and Simple Links of a Continuous Curve

Many of the definitions and results concerning connected im kleinen continua become useful only when these continua are locally compact. This is especially true in the cyclic element theory. For if a continuous curve M is not locally compact, it is not necessarily true that a simple closed curve in M belongs to a cyclic element of M. Furthermore, if a continuous curve M is not locally compact, it is not necessarily true that a simple closed curve in M belongs to a simple link in M. However, if I f is a continuous curve, there are in M certain subcontinua which strongly resemble both cyclic elements and simple links such that if J is a simple closed curve in M, one of them contains J. I t is the purpose of this paper to define these sets, to develop a few of their properties, and to show how they are of considerable interest in spaces where the Jordan curve theorem holds true.