Mathematics
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Item Completions(1964) Nielsen, Robert Maurice; Brace, John W.; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md)This paper presents a new approach to the theory of completions. The treatment is based on the concept of convergence on filters and related topologies. For a given uniform Hausdorff space Xu and a collection S of Cauchy filters in Xu, the basic result is the construction of a uniform Hausdorff space. Xu having the properties that Xu is isomorphic to a dense subspace of Xu and every filter in S converges to a point in S. As a special case, the completion of Xu of Xu is obtained. The construction is so given as to prove the existence of the space Xu. The technique involves embedding the object X to be "completed" in a space of functions F which has as its domain a space of continuous functions C(X) defined on X. The procedure is analogous to the process of taking the bidual E" of a locally convex topological vector space. Indeed, E" is obtained as a special case. In the absence of sufficient structure on X, the Xu is obtained as the closure of X in F. In a locally convex space or an abelian topological group having enough character to separate points, Xu is obtained as a bidual or a second character group of the object X.Item A COMBINATORIAL REPRESENTATION FOR ORIENTED POLYHEDRAL SURFACES(1960) Edmonds, John Robert Jr; Reinhart, Bruce; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md)Item The Axiom of Choice for Collections of Finite Sets(1969) Gauntt, Robert James; Karp, Carol R.; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md)Some implications among finite versions of the Axiom of Choice are considered. In the first of two chapters some theorems are proven concerning the dependence or independence of these implications on the theory ZFU, the modification of ZF which permits the existence of atoms. The second chapter outlines proofs of corresponding theorems with "ZFU" replaced by "ZF" . The independence proofs involve Mostowski type permutation models in the first chapter and Cohen forcing in the second chapter. The finite axioms considered are C^n , "Every collection of n-element sets has a choice function"; W^n, "Every well-orderable collection of n-element sets has a choice function"; D^n, "Every denumerable collection of n-element sets has a choice function"; and A^n (x), "Every collection Y of n-element sets, with Y ≈ X, has a choice function". The conjunction C^nl &...& C^nk is denoted by CZ where Z = {nl ,...,nk}. Corresponding conjunctions of other finite axioms are denoted similarly by Wz, Dz and Az (X). Theorem: The following are provable in ZFU: W^k1n1+...+krnr ➔ W^n1 v...v W^nr, D^k1n1+...+krnr ➔ D^n1 v...v D^nr, and C^k1n1+...+krnr ➔ C^n1 v W^n2 v...v W^nr