Inaug.-Diss.--Marburg. / Vita. Bibliography: 5th prelim. p.

Inaug.-diss.--Universität Bern. / "Literatur": p. [137]-38.

Academisch proefschrift. / Includes bibliographical references.

Thèse--Univ. de Paris.

Six definitions of a finite set are studied; and each implication between the definitions is shown to be either derivable from the Zermelo-Fraenkel axioms or independent of them. The method of Boolean-valued models, as studied by D. Scott, is used to show that it is consistent with the Zermelo-Fraenkel axioms to deny some of these implications by constructing appropriate models. The six definitions are shown to satisfy a list of properties which a definition of finite should reasonably possess. It is also shown that any definition which satisfies these properties must encompass a larger class of sets than the first definition and a smaller class than the sixth. Three more definitions, which do not possess these properties, are mentioned, and the differences between these definitions and the first six are discussed. / Science, Faculty of / Mathematics, Department of / Graduate

This thesis deals with some equivalences to Soulin's hypothesis. Also included is a consequence of its negation, the existence of a normal Hausdorff space which is not countably paracompact. Two equivalences to Souslin's hypothesis can be obtained by considering the countable chain condition in the product with itself of an ordered continuum and by considering Peano maps on ordered continua. An equivalence to a negative answer is the existence of a tree of cardinality, [formula omitted] ₁, in which every chain and antichain (set of pairwise incomparable objects) is countable. Another equivalence to Souslin's hypothesis is obtained by assuming that a continuum which is the continuous image of an ordered continuum is metrizable if and only if it has the countable chain condition. We get yet another by assuming that a compactum which is the continuous image of an ordered compactum is separable if and only if it has the countable chain condition. Finally, if we restrict ourselves to compacta of the type just mentioned we find that Souslin's hypothesis is equivalent to a generalization of a metrization theorem of S. Mardeśic. / Science, Faculty of / Mathematics, Department of / Graduate

No description available.

Thesis (Ph.D.)--Tufts University, 2001. / Adviser: Alva Couch. Submitted to the Dept. of Computer Science. Includes bibliographical references (leaves 144-148). Access restricted to members of the Tufts University community. Also available via the World Wide Web;

The purpose of this paper is to examine selected topologies, the Vietoris topology in particular, on S(X), the collection of nonempty, closed subsets of a topological space X. Characteristics of open and closed subsets of S(X), with the Vietoris topology, are noted. The relationships between the space X and the space S(X), with the Vietoris topology, concerning the properties of countability, compactness, and connectedness and the separation properties are investigated. Additional topologies are defined on S(X), and each is compared to the Vietoris topology on S(X). Finally, topological convergence of nets of subsets of X is considered. It is found that topological convergence induces a topology on S(X), and that this topology is the Vietoris topology on S(X) when X is a compact, Hausdorff space.

Let M be the class of simple matroids which do not contain the 5-point line U2,5 , the Fano plane F7 , the non-Fano plane F7- , or the matroid P7 , as minors. Let h(n) be the maximum number of points in a rank-n matroid in M. We show that h(2)=4, h(3)=7, and h(n)=n(n+1)/2 for n>3, and we also find all the maximum-sized matroids for each rank.