## Dissertations from ProQuest

#### Title

Span and real functional diameter of metric continua

1998

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.

Mathematics