Title

Span and real functional diameter of metric continua

Date of Award

1998

Availability

Article

Degree Name

Doctor of Philosophy (Ph.D.)

First Committee Member

Edwin Duda, Committee Chair

Abstract

In this dissertation we study the relation of the real functional diameter of a metric continuum (X,d), denoted by $D\sb{R}(X)$, and the span of such a continuum, denoted by $\sigma(X).$ Both $D\sb{R}(X)$ and $\sigma(X)$ are non-negative real numbers, and we include several examples for which we calculate $D\sb{R}(X)$ and $\sigma(X).$ In one example, we demonstrate that the real functional diameter of a planar simple closed curve J need not be equal to the real functional diameter of the continuum formed by taking the union of J with the bounded component of the complement of J in the plane.We demonstrate a method for extending a continuous function from the boundary of a compact convex regular body, in the plane, to its interior in such a manner as not to increase the diameter of point inverses. We apply this result in proving that the real functional diameter of a compact convex planar set is equal to the real functional diameter of the boundary of such a set. In relation to planar convex sets, we apply results from the theory of multi-valued functions in showing that the span and real functional diameter of compact convex planar sets are equal.We demonstrate that the real functional diameter of a metric continuum is an upper bound for the span of such a continuum. Moreover, for plane continua we obtain an upper bound for the span of such a continuum by using translations of the plane. We also obtain a lower bound for the span of a plane separating continuum X. In particular, we show that if a convex body is contained in a bounded component of the complement of X in the plane, the span of the boundary of the convex set is a lower bound for the span of X.

Keywords

Mathematics

Link to Full Text

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