Date of Award
Master of Science (MS)
Mathematics and Statistics
L. D. Loveland
L. D. Loveland
In the course of studying continua in the plane it has been asked if a given continuum has uncountably many disjoint duplications in the plane, and if so, what are the consequences of the existence of such a collection. The object of this paper is to study these problems and to develop some machinery useful in their resolution. In Section I, we review the definition of convergence and homeomorphic convergence of point sets in a metric space S. We then consider the space, π of all continuous functions from a compact metric space P to a separable metric space Q under the "sup metric." The famous Borsuk Theorem , which shows that π is separable, is then proven. As a consequence of the separability of π , we show that in any uncountable collect ion G of compact sets in a separable metric space S, there must exist an element of G that is converged to by a sequence of elements of G. Furthermore, if the elements of Gare pairwise homeomorphic, then G contains an element which is converged to homeomorphically by some sequence in G. These theorems prove to be important in the study of disjoint embeddings of continua in separable metric spaces. In Section I, as throughout this paper, definitions are motivated and illustrated by numerous examples.
Matthews, Steven Ray, "Classifications of Plane Continua" (1972). All Graduate Plan B and other Reports. 1173.
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