Harold Reiter

Separating Function Algebras

Recent results of Hoffman and Singer [7], Weiss [10] and Wilken [11] indicate that the study of separation properties play a central role in the theory of function algebras. Our purpose in this paper is to investigate a natural separation property of function algebras.

The Space of Closed Subsets of a Convergent Sequence

Many topological spaces are simply sets of points (atoms) endowed with a topology. Some spaces, however, have elements that are functions, matrices, or other non-atomic items. Another special type of space has elements that are themselves subsets of another space; these spaces are called hyperspaces. Hyperspaces are metric spaces, and the metric defined on them is called the Hausdorff metric. Hyperspaces wllose points are the closed subsets and hyperspaces whose points are the closed connected subsets (of metric spaces) have been extensively studied. Also, hyperspaces of closed and convex subsets of a bounded convex set in Euclidean space are of great interest in geometry. See Lay [3]. In recent years, geometers and topological dynamicists have explored spaces of closed and bounded subsets of the plane in connection with the study of fractals. One of the major results in the theory is that the hyperspace of closed subsets of a closed interval of real numbers is homeomorphic with the Hilbert cube, and with the space IX, the countable product of unit intervals. For more general information about the Hausdorff metric and spaces of subsets, see Devaney [2] and Sieradski [6]. The main result of this paper is a topological characterization of the space of closed subsets of a convergent sequence of points. The proof given here provides a homneornorphic embedding of the space in the plane, E2. The result was first proved by Pelczynski [5]. He studied arbitrary compact zero-dimensional metric spaces, so his proofs are much more widely applicable, but they are also somewhat technical. Our proof depends only on some well-known results in the theory of metric spaces, and is therefore accessible to advanced undergraduate mathematics majors. We also prove that for no convergent sequence of real numbers is there an isometric embedding of the hyperspace in Euclidean space, E,1, for any n. For other results and discussion, see Nadler [4]. Let (X, d) denote a compact metric space. The hyperspace (2X, D) of X is the metric space whose points are the closed nonempty subsets of X and whose metric is the Hausdorff metric D given by

The "Join the Club" Interpretation of Some Graph Algorithms

Several important tree construction algorithms of graph theory are described and discussed using an easily remembered interpretation. Under this interpretation, vertices are thought of as potential members of a club. A rule for adjoining new vertices to the tree corresponds to a specific rule by which new members join the club. Interest in graph theory has accelerated during the last two decades because of its far-reaching applications in other branches of mathematics, especially operations research, as well as outside of mathematics. There is hardly a discipline cannot lay claim to advances attained by the development of methods and models of graph theory. Computer scientists and economists use communication and transportation networks, sociologists employ graphs to study social mobility and stratification, anthropologists use interval graphs in the dating of artifacts, psychologists use them in clustering and in transactional analysis, and chemists count and study isomers using graphs and tree models. The abundance of “real life” situations for which graph theory is useful has encouraged many mathematicians and non-mathematicians to learn and use it. An important part of graph theory is concerned with algorithms. Searching and sorting, finding shortest paths, constructing spanning trees with desirable properties, matching and coloring vertices are problems whose solutions are algorithms. Hence, nearly every course in discrete mathematics, combinatorics, and graph theory contains some material on algorithms. Some students find graph algorithms hard to remember. Though most algorithms are based on relatively simple ideas, their formal presentation may be rather long, and understanding them may require substantial mathematical maturity. Another difficulty students sometimes encounter is that often algorithms

A Complete Solution to the Magic Hexagram Problem

Harold Reiter earned the B.S. degree at LSU in 1964 and the Ph.D. at Clemson in 1969. Except for three year stints at Hawaii (Manoa) and at Maryland (College Park), the latter in computer science, he has been at UNC Charlotte since 1972. His professional interests include recreational mathematics and teacher education. His recreational interests include tennis and racketball, jogging, reading, and travel. Also, he enjoys driving his sports car with license plate CALCULUS. David Ritchie was born and raised in Gambrills, Maryland, which boasts of a grocery store as well as a post office. He had an average childhood, but became engrossed by computers sometime in the tenth grade. More recently, he graduated from the University of Maryland with a B.S. in computer science. When not otherwise engaged with his wife or his computer, he finds time to play racketball and enjoy some water-skiing.