Notions of closed subsets of a complete separable metric space in weak subsystems of second-order arithmetic

1990 ◽  
pp. 39-50 ◽  
Author(s):  
Douglas K. Brown
Keyword(s):  
Metric Space ◽  
Second Order ◽  
Second Order Arithmetic ◽  
Order Arithmetic ◽  
Complete Separable Metric Space ◽  
Separable Metric Space

Reverse Mathematics and Π12 Comprehension

2005 ◽  
Vol 11 (4) ◽  
pp. 526-533 ◽  
Author(s):  
Carl Mummert ◽  
Stephen G. Simpson
Keyword(s):  
Topological Space ◽  
Metric Space ◽  
Reverse Mathematics ◽  
Second Order ◽  
Converse Statement ◽  
Second Order Arithmetic ◽  
Order Arithmetic ◽  
Complete Separable Metric Space ◽  
Separable Metric Space ◽  
Metrization Theorem

AbstractWe initiate the reverse mathematics of general topology. We show that a certain metrization theorem is equivalent to Π12 comprehension. An MF space is defined to be a topological space of the form MF(P) with the topology generated by {Np ∣ p ϵ P}. Here P is a poset, MF(P) is the set of maximal filters on P, and Np = {F ϵ MF(P) ∣ p ϵ F }. If the poset P is countable, the space MF(P) is said to be countably based. The class of countably based MF spaces can be defined and discussed within the subsystem ACA0 of second order arithmetic. One can prove within ACA0 that every complete separable metric space is homeomorphic to a countably based MF space which is regular. We show that the converse statement, “every countably based MF space which is regular is homeomorphic to a complete separable metric space,” is equivalent to . The equivalence is proved in the weaker system . This is the first example of a theorem of core mathematics which is provable in second order arithmetic and implies Π12 comprehension.

REVERSE MATHEMATICS OF MF SPACES

2006 ◽  
Vol 06 (02) ◽  
pp. 203-232 ◽  
Author(s):  
CARL MUMMERT
Keyword(s):  
Metric Space ◽  
Closed Subset ◽  
Metric Spaces ◽  
Reverse Mathematics ◽  
General Topology ◽  
Second Order Arithmetic ◽  
Perfect Set ◽  
Order Arithmetic ◽  
Complete Separable Metric Space ◽  
Separable Metric Space

This paper gives a formalization of general topology in second-order arithmetic using countably based MF spaces. This formalization is used to study the reverse mathematics of general topology. For each poset P we let MF (P) denote the set of maximal filters on P endowed with the topology generated by {Np | p ∈ P}, where Np = {F ∈ MF (P) | p ∈ F}. We define a countably based MF space to be a space of the form MF (P) for some countable poset P. The class of countably based MF spaces includes all complete separable metric spaces as well as many nonmetrizable spaces. The following reverse mathematics results are obtained. The proposition that every nonempty Gδ subset of a countably based MF space is homeomorphic to a countably based MF space is equivalent to [Formula: see text] over ACA0. The proposition that every uncountable closed subset of a countably based MF space contains a perfect set is equivalent over [Formula: see text] to the proposition that [Formula: see text] is countable for all A ⊆ ℕ. The proposition that every regular countably based MF space is homeomorphic to a complete separable metric space is equivalent to [Formula: see text] over [Formula: see text].

Located sets and reverse mathematics

2000 ◽  
Vol 65 (3) ◽  
pp. 1451-1480 ◽  
Author(s):  
Mariagnese Giusto ◽  
Stephen G. Simpson
Keyword(s):  
Distance Function ◽  
Metric Space ◽  
Extension Theorem ◽  
Reverse Mathematics ◽  
Compact Metric Space ◽  
Closed Set ◽  
Second Order Arithmetic ◽  
Closed Sets ◽  
Order Arithmetic ◽  
Separable Metric Space

AbstractLet X be a compact metric space. A closed set K ⊆ X is located if the distance function d(x, K) exists as a continuous real-valued function on X; weakly located if the predicate d(x, K) > r is allowing parameters. The purpose of this paper is to explore the concepts of located and weakly located subsets of a compact separable metric space in the context of subsystems of second order arithmetic such as RCA0, WKL0 and ACA0. We also give some applications of these concepts by discussing some versions of the Tietze extension theorem. In particular we prove an RCA0 version of this result for weakly located closed sets.

Logic and computation, Proceedings of a workshop held at Carnegie Mellon University, June 30–July 2, 1987, edited by Wilfried Sieg, Contemporary Mathematics, vol. 106, American Mathematical Society, Providence1990, xiv + 297 pp. - Douglas K. Brown. Notions of closed subsets of a complete separable metric space in weak subsystems of second order arithmetic. Pp. 39–50. - Kostas Hatzikiriakou and Stephen G. Simpson. WKL0 and orderings of countable abelian groups. Pp. 177–180. - Jeffry L. Hirst. Marriage theorems and reverse mathematics. Pp. 181–196. - Xiaokang Yu. Radon–Nikodym theorem is equivalent to arithmetical comprehension. Pp. 289–297. - Fernando Ferreira. Polynomial time computable arithmetic. Pp. 137–156. - Wilfried Buchholz and Wilfried Sieg. A note on polynomial time computable arithmetic. Pp. 51–55. - Samuel R. Buss. Axiomatizations and conservation results for fragments of bounded arithmetic. Pp. 57–84. - Gaisi Takeuti. Sharply bounded arithmetic and the function a – 1. Pp. 281–288. - Peter G. Clote. A smash-based hierarchy between PTIME and PSPACE (preliminary version). Pp. 85–100. - Solomon Feferman. Polymorphic typed lambda-calculi in a type-free axiomatic framework. Pp. 101–136 - Michael Beeson. Some theories conservative over intuitionistic arithmetic. Pp. 1–15.

1996 ◽  
Vol 61 (2) ◽  
pp. 697-699
Author(s):  
Jörg Hudelmaier
Keyword(s):  
Polynomial Time ◽  
Metric Space ◽  
American Mathematical Society ◽  
Bounded Arithmetic ◽  
Preliminary Version ◽  
Order Arithmetic ◽  
Complete Separable Metric Space ◽  
Separable Metric Space ◽  
Intuitionistic Arithmetic ◽  
Nikodym Theorem

scholarly journals Short note: Least fixed points versus least closed points

2021 ◽  
Author(s):  
Gerhard Jäger
Keyword(s):  
Fixed Points ◽  
Short Note ◽  
Second Order ◽  
Second Order Arithmetic ◽  
Order Arithmetic ◽  
Arithmetic Operator

AbstractThis short note is on the question whether the intersection of all fixed points of a positive arithmetic operator and the intersection of all its closed points can proved to be equivalent in a weak fragment of second order arithmetic.

UNDECIDABILITY OF THE THEORIES OF CLASSES OF STRUCTURES

2014 ◽  
Vol 79 (4) ◽  
pp. 1001-1019 ◽  
Author(s):  
ASHER M. KACH ◽  
ANTONIO MONTALBÁN
Keyword(s):  
Direct Product ◽  
Disjoint Union ◽  
Second Order ◽  
Linear Orders ◽  
Second Order Arithmetic ◽  
Order Arithmetic ◽  
Natural Classes ◽  
Product Of Groups

AbstractMany classes of structures have natural functions and relations on them: concatenation of linear orders, direct product of groups, disjoint union of equivalence structures, and so on. Here, we study the (un)decidability of the theory of several natural classes of structures with appropriate functions and relations. For some of these classes of structures, the resulting theory is decidable; for some of these classes of structures, the resulting theory is bi-interpretable with second-order arithmetic.

scholarly journals On limit theorems for random fields

2009 ◽  
Vol 50 ◽  
Author(s):  
Rimas Banys
Keyword(s):  
Metric Space ◽  
Random Fields ◽  
Limit Theorems ◽  
Characteristic Property ◽  
Complete Separable Metric Space ◽  
Separable Metric Space

A complete separable metric space of functions defined on the positive quadrant of the plane is constructed. The characteristic property of these functions is that at every point x there exist two lines intersecting at this point such that limits limy→x f (y) exist when y approaches x along any path not intersecting these lines. A criterion of compactness of subsets of this space is obtained.

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