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].

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.

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 Uniform Spaces as Nice Images of Nice Uniform and Metric Spaces(1)

1980 ◽  
Vol 23 (1) ◽  
pp. 1-9
Author(s):  
Richard Willmott
Keyword(s):  
Metric Space ◽  
Closed Subset ◽  
Metric Spaces ◽  
General Question ◽  
Uniform Spaces ◽  
Continuous Mappings ◽  
Analytic Sets ◽  
One To One ◽  
Complete Separable Metric Space ◽  
Descriptive Set

The classical theorem that a complete separable metric space is the image under a one-to-one continuous function of a closed subset of the irrational numbers has been extended in two directions, the first leading to various characterizations in descriptive set theory of Borel and analytic sets or generalizations of them as continuous images of certain subsets of the irrationals, or generalizations of them (see, e.g. [3] and references cited there; [4]; [6]). The second direction originates in the observation that a closed subset of the irrationals is a complete 0-dimensional metric space (under a suitable metric), and leads to the general question asked by Alexandroff [1], "Which spaces can be represented as images of 'nice' (e.g. metric, 0-dimensional) spaces under 'nice' [e.g. one-to-one, open, closed, perfect] continuous mappings?" (See, e.g. [7], [9] and the survey articles [1], [2] and [11].)

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.

Reverse Mathematics: The Playground of Logic

2010 ◽  
Vol 16 (3) ◽  
pp. 378-402 ◽  
Author(s):  
Richard A. Shore
Keyword(s):  
Mathematical Logic ◽  
Reverse Mathematics ◽  
Computational Approach ◽  
Second Order ◽  
Second Order Arithmetic ◽  
Base Theory ◽  
Order Arithmetic

AbstractThis paper is essentially the author's Gödel Lecture at the ASL Logic Colloquium '09 in Sofia extended and supplemented by material from some other papers. After a brief description of traditional reverse mathematics, a computational approach to is presented. There are then discussions of some interactions between reverse mathematics and the major branches of mathematical logic in terms of the techniques they supply as well as theorems for analysis. The emphasis here is on ones that lie outside the usual main systems of reverse mathematics. While retaining the usual base theory and working still within second order arithmetic, theorems are described that range from those far below the usual systems to ones far above.

scholarly journals A Universal Separable Diversity

2017 ◽  
Vol 5 (1) ◽  
pp. 138-151 ◽  
Author(s):  
David Bryant ◽  
André Nies ◽  
Paul Tupper
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Complete Metric Space ◽  
Urysohn Space ◽  
Fundamental Properties ◽  
Finite Set ◽  
Whole Space ◽  
Set Of Points ◽  
Separable Metric Space

AbstractThe Urysohn space is a separable complete metric space with two fundamental properties: (a) universality: every separable metric space can be isometrically embedded in it; (b) ultrahomogeneity: every finite isometry between two finite subspaces can be extended to an auto-isometry of the whole space. The Urysohn space is uniquely determined up to isometry within separable metric spaces by these two properties. We introduce an analogue of the Urysohn space for diversities, a recently developed variant of the concept of a metric space. In a diversity any finite set of points is assigned a non-negative value, extending the notion of a metric which only applies to unordered pairs of points.We construct the unique separable complete diversity that it is ultrahomogeneous and universal with respect to separable diversities.

Weak-open images of locally separable metric spaces

2011 ◽  
Vol 48 (2) ◽  
pp. 145-159
Author(s):  
Zhaowen Li ◽  
Xun Ge ◽  
Qingguo Li
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Weak Base ◽  
Cosmic Space ◽  
Locally Separable Metric Spaces ◽  
Separable Metric Space

In this paper, we prove that a space X is a weak-open compact image of a locally separable metric space if and only if X has a uniform cosmic-weak-base if and only if X is a weak-open compact image of a metric space and a locally cosmic space, and give some internal characterizations of weak-open s-images of locally separable metric spaces.

Martin's axiom and Hausdorif measures

1974 ◽  
Vol 75 (2) ◽  
pp. 193-197 ◽  
Author(s):  
A. J. Ostaszewski
Keyword(s):  
Metric Space ◽  
Measure Zero ◽  
Continuum Hypothesis ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Complete Separable Metric Space ◽  
Separable Metric Space ◽  
The Continuum

AbstractA theorem of Besicovitch, namely that, assuming the continuum hypothesis, there exists in any uncountable complete separable metric space a set of cardinality the continuum all of whose Hausdorif h-measures are zero, is here deduced by appeal to Martin's Axiom. It is also shown that for measures λ of Hausdorff type the union of fewer than 2ℵ0 sets of λ-measure zero is also of λ-measure zero; furthermore, the union of fewer than 2ℵ0 λ-measurable sets is λ-measurable.

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