Concerning upper semi-continuous collections of continua

Abstract

Hurewicz [1](2) and Mazurkiewicz [2] showed independently that if M is any compact metric continuum, there exist a one-dimensional continuum K in three-dimensional Euclidean space and an upper semi-continuous collection [3 ] of mutually exclusive continua filling up K which with respect to its elements as points is topologically equivalent to M. In the case of each of these solutions the method of proof used does not lend itself readily to the solution of the principal result of this paper which is the demonstration that if M is any compact continuous curve, there exist a one-dimensional continuous curve K in three-dimensional Euclidean space and an upper semi-continuous collection of mutually exclusive continua filling up K which with respect to its elements as points is topologically equivalent to M. It is known that, in the problem of Hurewicz and Mazurkiewicz, if M is not a continuous curve then K cannot be a continuous curve. The principal result in this paper, Theorem II, was proposed to me as a problem by Professor R. L. Moore. I wish to express my sincere appreciation to Professor Moore for his patient and stimulating teaching and for his contagious enthusiasm for mathematical research. DEFINITION. A collection Q of continuous curves will be said to have the X property if the common part of the continua of any subcollection of Q has only a finite number of components, each a continuous curve.