FORCING, GAMES and FAMILIES of CLOSED SETS 1. Introduction
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Calibrating Determinacy Strength in Levels of the Borel Hierarchy
CALIBRATING DETERMINACY STRENGTH IN LEVELS OF THE BOREL HIERARCHY SHERWOOD J. HACHTMAN Abstract. We analyze the set-theoretic strength of determinacy for levels of the Borel 0 hierarchy of the form Σ1+α+3, for α < !1. Well-known results of H. Friedman and D.A. Martin have shown this determinacy to require α+1 iterations of the Power Set Axiom, but we ask what additional ambient set theory is strictly necessary. To this end, we isolate a family of Π1-reflection principles, Π1-RAPα, whose consistency strength corresponds 0 CK exactly to that of Σ1+α+3-Determinacy, for α < !1 . This yields a characterization of the levels of L by or at which winning strategies in these games must be constructed. When α = 0, we have the following concise result: the least θ so that all winning strategies 0 in Σ4 games belong to Lθ+1 is the least so that Lθ j= \P(!) exists + all wellfounded trees are ranked". x1. Introduction. Given a set A ⊆ !! of sequences of natural numbers, consider a game, G(A), where two players, I and II, take turns picking elements of a sequence hx0; x1; x2;::: i of naturals. Player I wins the game if the sequence obtained belongs to A; otherwise, II wins. For a collection Γ of subsets of !!, Γ determinacy, which we abbreviate Γ-DET, is the statement that for every A 2 Γ, one of the players has a winning strategy in G(A). It is a much-studied phenomenon that Γ -DET has mathematical strength: the bigger the pointclass Γ, the stronger the theory required to prove Γ -DET. -
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382 Title: Regularized Set Operations in Solid Modeling Revisited Authors: K. M. Yu, [email protected], The Hong Kong Polytechnic University K. M. Lee, [email protected], The Hong Kong Polytechnic University K. M. Au, [email protected], HKU SPACE Community College Keywords: Regularised Set Operations, CSG, Solid Modeling, Manufacturing, General Topology DOI: 10.14733/cadconfP.2019.382-386 Introduction: The seminal works of Tilove and Requicha [8],[13],[14] first introduced the use of general topology concepts in solid modeling. However, their works are brute-force approach from mathematicians’ perspective and are not easy to be comprehended by engineering trained personnels. This paper reviewed the point set topological approach using operational formulation. Essential topological concepts are described and visualized in easy to understand directed graphs. These facilitate subtle differentiation of key concepts to be recognized. In particular, some results are restated in a more mathematically correct version. Examples to use the prefix unary operators are demonstrated in solid modeling properties derivations and proofs as well as engineering applications. Main Idea: This paper is motivated by confusion in the use of mathematical terms like boundedness, boundary, interior, open set, exterior, complement, closed set, closure, regular set, etc. Hasse diagrams are drawn by mathematicians to study different topologies. Instead, directed graph using elementary operators of closure and complement are drawn to show their inter-relationship in the Euclidean three- dimensional space (E^3 is a special metric space formed from Cartesian product R^3 with Euclid’s postulates [11] and inheriting all general topology properties.) In mathematics, regular, regular open and regular closed are similar but different concepts. -
Effective Cardinals of Boldface Pointclasses
EFFECTIVE CARDINALS OF BOLDFACE POINTCLASSES ALESSANDRO ANDRETTA, GREG HJORTH, AND ITAY NEEMAN Abstract. Assuming AD + DC(R), we characterize the self-dual boldface point- classes which are strictly larger (in terms of cardinality) than the pointclasses contained in them: these are exactly the clopen sets, the collections of all sets of ξ ξ Wadge rank ≤ ω1, and those of Wadge rank < ω1 when ξ is limit. 1. Introduction A boldface pointclass (for short: a pointclass) is a non-empty collection Γ of subsets of R such that Γ is closed under continuous pre-images and Γ 6= P(R). 0 0 0 Examples of pointclasses are the levels Σα, Πα, ∆α of the Borel hierarchy and the 1 1 1 levels Σn, Πn, ∆n of the projective hierarchy. In this paper we address the following Question 1. What is the cardinality of a pointclass? Assuming AC, the Axiom of Choice, Question 1 becomes trivial for all pointclasses Γ which admit a complete set. These pointclasses all have size 2ℵ0 under AC. On the other hand there is no obvious, natural way to associate, in a one-to-one way, 0 an open set (or for that matter: a closed set, or a real number) to any Σ2 set. This 0 0 suggests that in the realm of definable sets and functions already Σ1 and Σ2 may have different sizes. Indeed the second author in [Hjo98] and [Hjo02] showed that AD + V = L(R) actually implies 0 0 (a) 1 ≤ α < β < ω1 =⇒ |Σα| < |Σβ|, and 1 1 1 (b) |∆1| < |Σ1| < |Σ2| < . -
A Concise Form of Continuity in Fine Topological Space
Advances in Computational Sciences and Technology (ACST). ISSN 0973-6107 Volume 10, Number 6 (2017), pp. 1785–1805 © Research India Publications http://www.ripublication.com/acst.htm A Concise form of Continuity in Fine Topological Space P.L. Powar and Pratibha Dubey Department of Mathematics and Computer Science, R.D.V.V., Jabalpur, India. Abstract By using the topology on a space X, a wide class of sets called fine open sets have been studied earlier. In the present paper, it has been noticed and verified that the class of fine open sets contains the entire class of A-sets, AC-sets and αAB-sets, which are already defined. Further, this observation leads to define a more general continuous function which in tern reduces to the four continuous functions namely A-continuity, AB-continuity, AC-continuity and αAB-continuity as special cases. AMS subject classification: primary54XX; secondry 54CXX. Keywords: fine αAB-sets, fine A-sets, fine topology. 1. Introduction The concept of fine space has been introduced by Powar and Rajak in [1] is the special case of generalized topological space (see[2]). The major difference between the two spaces is that the open sets in the fine space are not the random collection of subsets of X satisfying certain conditions but the open sets (known as fine sets) have been generated with the help of topology already defined over X. It has been studied in [1] that this collection of fine open sets is really a magical class of subsets of X containing semi-open sets, pre-open sets, α-open sets, β-open sets etc. -
The Envelope of a Pointclass Under a Local Determinacy Hypothesis
THE ENVELOPE OF A POINTCLASS UNDER A LOCAL DETERMINACY HYPOTHESIS TREVOR M. WILSON Abstract. Given an inductive-like pointclass Γ and assuming the Ax- iom of Determinacy, Martin identified and analyzede the pointclass con- taining the norm relations of the next semiscale beyond Γ, if one exists. We show that much of Martin's analysis can be carriede out assuming only ZF + DCR + Det(∆Γ). This generalization requires arguments from Kechris{Woodin [10] ande e Martin [13]. The results of [10] and [13] can then be recovered as immediate corollaries of the general analysis. We also obtain a new proof of a theorem of Woodin on divergent models of AD+, as well as a new result regarding the derived model at an inde- structibly weakly compact limit of Woodin cardinals. Introduction Given an inductive-like pointclass Γ, Martin introduced a pointclass (which we call the envelope of Γ, following [22])e whose main feature is that it con- tains the prewellorderingse of the next scale (or semiscale) beyond Γ, if such a (semi)scale exists. This work was distributed as \Notes on the nexte Suslin cardinal," as cited in Jackson [3]. It is unpublished, so we use [3] as our reference instead for several of Martin's arguments. Martin's analysis used the assumption of the Axiom of Determinacy. We reformulate the notion of the envelope in such a way that many of its essen- tial properties can by derived without assuming AD. Instead we will work in the base theory ZF+DCR for the duration of the paper, stating determinacy hypotheses only when necessary. -
Arxiv:1710.00393V2
DIMENSION, COMPARISON, AND ALMOST FINITENESS DAVID KERR Abstract. We develop a dynamical version of some of the theory surrounding the Toms–Winter conjecture for simple separable nuclear C∗-algebras and study its con- nections to the C∗-algebra side via the crossed product. We introduce an analogue of hyperfiniteness for free actions of amenable groups on compact spaces and show that it plays the role of Z-stability in the Toms–Winter conjecture in its relation to dynamical comparison, and also that it implies Z-stability of the crossed product. This property, which we call almost finiteness, generalizes Matui’s notion of the same name from the zero-dimensional setting. We also introduce a notion of tower dimen- sion as a partial analogue of nuclear dimension and study its relation to dynamical comparison and almost finiteness, as well as to the dynamical asymptotic dimension and amenability dimension of Guentner, Willett, and Yu. 1. Introduction Two of the cornerstones of the theory of von Neumann algebras with separable predual are the following theorems due to Murray–von Neumann [31] and Connes [5], respectively: (i) there is a unique hyperfinite II1 factor, (ii) injectivity is equivalent to hyperfiniteness. Injectivity is a form of amenability that gives operator-algebraic expression to the idea of having an invariant mean, while hyperfiniteness means that the algebra can be expressed as the weak operator closure of an increasing sequence of finite-dimensional ∗-subalgebras (or, equivalently, that one has local ∗-ultrastrong approximation by such ∗-subalgebras [12]). The basic prototype for the relation between an invariant-mean- arXiv:1710.00393v2 [math.DS] 13 Jan 2019 type property and finite or finite-dimensional approximation is the equivalence between amenability and the Følner property for discrete groups, and indeed Connes’s proof of (ii) draws part of its inspiration from the Day–Namioka proof of this equivalence. -
Determinacy and Large Cardinals
Determinacy and Large Cardinals Itay Neeman∗ Abstract. The principle of determinacy has been crucial to the study of definable sets of real numbers. This paper surveys some of the uses of determinacy, concentrating specifically on the connection between determinacy and large cardinals, and takes this connection further, to the level of games of length ω1. Mathematics Subject Classification (2000). 03E55; 03E60; 03E45; 03E15. Keywords. Determinacy, iteration trees, large cardinals, long games, Woodin cardinals. 1. Determinacy Let ωω denote the set of infinite sequences of natural numbers. For A ⊂ ωω let Gω(A) denote the length ω game with payoff A. The format of Gω(A) is displayed in Diagram 1. Two players, denoted I and II, alternate playing natural numbers forming together a sequence x = hx(n) | n < ωi in ωω called a run of the game. The run is won by player I if x ∈ A, and otherwise the run is won by player II. I x(0) x(2) ...... II x(1) x(3) ...... Diagram 1. The game Gω(A). A game is determined if one of the players has a winning strategy. The set A is ω determined if Gω(A) is determined. For Γ ⊂ P(ω ), det(Γ) denotes the statement that all sets in Γ are determined. Using the axiom of choice, or more specifically using a wellordering of the reals, it is easy to construct a non-determined set A. det(P(ωω)) is therefore false. On the other hand it has become clear through research over the years that det(Γ) is true if all the sets in Γ are definable by some concrete means. -
MS Word Template for CAD Conference Papers
1084 Regularized Set Operations in Solid Modeling Revisited K. M. Yu1 , K. M. Lee2 and K. M. Au 3 1 The Hong Kong Polytechnic University, [email protected] 2 The Hong Kong Polytechnic University, [email protected] 3 HKU SPACE Community College, [email protected] Corresponding author: K. M. Au, [email protected] Abstract. The seminal works of Tilove and Requicha first introduced the use of general topology concepts in solid modeling. However, their works are brute-force approach from mathematicians’ perspective and are not easy to be comprehended by engineering trained personnels. This paper reviewed the point set topological approach using operational formulation and provided an alternate method to check the correctness of regularized set operations that was difficult to be formulated or ignored in various research work or literature study of geometric modeling. The results simply provide an alternate approach that easier to check the correctness for future research and development in geometric modeling. Essential topological concepts are described and visualized in easy to understand directed graphs. These facilitate subtle differentiation of key concepts to be recognized. In particular, some results are restated in a more mathematically correct version. Examples to use the prefix unary operators are demonstrated in solid modeling properties derivations and proofs as well as engineering applications. Keywords: Regularised Set Operations, Constructive Solid Geometry, Solid Modeling, Manufacturing, General Topology. DOI: https://doi.org/10.14733/cadaps.2020.1084-1100 1 INTRODUCTION The seminal works of Tilove and Requicha [9],[14],[15] first introduced the use of general topology concepts in solid modeling. -
On Maximal and Minimal Μ-Clopen Sets in GT Spaces Rebati Mohan Roy1*
Malaya Journal of Matematik, Vol. 6, No. 4, 854-857 2018 https://doi.org/10.26637/MJM0604/0023 On maximal and minimal m-clopen sets in GT spaces Rebati Mohan Roy1* Abstract In this paper, we introduce the notions of maximal and minimal m-clopen sets in a generalized topological space and their some properties. We obtain that maximal and minimal m-clopen sets are independent of maximal and minimal m-open and m-closed sets. We observed that the existence of maximal m-clopen set in a generalized topological space not only ensure the m-disconnectedness of a generalized topological space but also the existence of minimal m-clopen set in that space. Keywords m-open set, m-closed set, maximal m-open set, minimal m-closed set, m-clopen, maximal m-clopen set, minimal m-clopen set. AMS Subject Classification 54A05, 54D05. 1Department of Mathematics, Mathabhanga College, Cooch Behar-736146, West Bengal, India. *Corresponding author: 1roy [email protected] Article History: Received 22 September 2018; Accepted 14 December 2018 c 2018 MJM. Contents denoted by im (A). The generalized closure of a subset A of X is the intersection of all m-closed sets containing A and is 1 Introduction.......................................854 denoted by cm (A). It is easy to see that im (A) = X −cm (X −A). 2 Maximal, minimal m-open set(resp. m-closed set) and By a proper m-open set (resp. m-closed set) of X, we mean m-clopen set ......................................854 a m-open set G (resp. m-closed set E) such that G 6= /0 and 3 Maximal and minimal m-clopen sets . -
0716-0917-Proy-40-03-671.Pdf
672 Ennis Rosas and Sarhad F. Namiq 1. Introduction Following [3] N. Levine, 1963, defined semi open sets. Similarly, S. F. Namiq [4], definedanoperationλ on the family of semi open sets in a topological space called semi operation, denoted by s-operation; via this operation, in his study [7], he defined λsc-open set by using λ-open and semi closed sets, and also following [5], he defined λco-open set and investigated several properties of λco-derived, λco-interior and λco-closure points in topological spaces. In the present article, we define the λco-connected space, discuss some characterizations and properties of λco-connected spaces, λco-components and λco-locally connected spaces and finally its relations with others con- nected spaces. 2. Preliminaries In the entire parts of the present paper, a topological space is referred to by (X, τ)orsimplybyX.First,somedefinitions are recalled and results are used in this paper. For any subset A of X, the closure and the interior of A are denoted by Cl(A)andInt(A), respectively. Following [8], the researchers state that a subset A of X is regular closed if A =Cl(Int(A)). Similarly, following [3], a subset A of a space X is semi open if A Cl(Int(A)). The complement of a semi open set is called semi closed. The⊆ family of all semi open (resp. semi closed) sets in a space X is denoted by SO(X, τ)orSO(X) (resp. SC(X, τ)orSC(X). According to [1], a space X is stated to be s- connected, if it is not the union of two nonempty disjoint semi open subsets of X.Weconsiderλ:SO(X) P (X) as a function definedonSO(X)into the power set of X, P (X)and→ λ is called a semi-operation denoted by s-operation, if V λ(V ), for each semi open set V . -
Compact Covering and Game Determinacy
&g-l.& __ TOPOLOGY I-- AND ITS EJ APPLKATIONS ELSEWIER Topology and its Applications 68 (1996) 153-185 Compact covering and game determinacy Gabriel Debs *, Jean Saint Raymond Equip d’Anulye, UnivemirP Puris 6. Boite 186, 4 pluce Jussieu, F-75252, Prtris Cedex 05, France Received 2 December 1994; revised 13 February 1995, 7 June 1995 Abstract All spaces are separable and metrizable. Suppose that the continuous and onto mapping f : X 4 Y is compact covering. Under the axiom of xi-determinacy, we prove that f is inductively perfect whenever X is Bore], and it follows then that Y is also Bore]. Under the axiom Nf = NI we construct examples showing that the conclusion might fail if “X is Borel” is replaced by “X is coanalytic”. If we suppose that both X and Y are Bore], then we prove (in ZFC) the weaker conclusion that f has a Bore1 (in fact a Baire-1) section g : Y + X. We also prove (in ZFC) that if we suppose only X to be Bore1 but of some “low” class, then Y is also Bore1 of the same class. Other related problems are discussed. Keyworcls: Compact covering; Inductively perfect; Determinacy AMS classification: 03E60; 04A15; 54E40; 54H05 Everywhere in this paper f : X -+ Y denotes a continuous and onto mapping between the spaces X and Y. By space we always mean a separable and metrizable space. We are interested in the comparison of the following two properties for f: (CC): “V L compact c Y, 3 K compact c X such that f(K) = L” and (IP): “3 X’ c X such that f(X’) = Y and V L compact c Y, the set X’ n f -’ (L) is compact”. -
A Classification of Projective-Like Hierarchies in L(R)
A classification of projective-like hierarchies in L(R) Rachid Atmai Universit¨atWien Kurt G¨odelResearch Center W¨ahringerStraße 25 1090 Vienna [email protected] Steve Jackson Dept. of Mathematics General Academics Building 435 1155 Union Circle 311430 University of North Texas Denton, TX 76203-5017 [email protected] October 10, 2016 Abstract We provide a characterization of projective-like hierarchies in L(R) in the context of AD by combinatorial properties of their associated Wadge ordinal. In the second chapter we concentrate on the type III case of the analysis. In the last chapter, we provide a characterization of type IV projective-like hierarchies by their associated Wadge ordinal. 1 Introduction 1.1 Outline The paper is roughly organized as follows. We first review basic definitions of the abstract theory of pointclasses. In the second section we study the type III pointclasses and the clo- sure properties of the Steel pointclass generated by a projective algebra Λ in the case where cof(o(Λ)) > !. In particular we show that Steel's conjecture is false. This leads to isolat- ing a combinatorial property of the Wadge ordinal of the projective algebra generating the 1 Steel pointclass. This combinatorial property ensures closure of the Steel pointclass under dis- junctions. Therefore it is still possible to have κ regular, where κ is the Wadge ordinal for a projective algebra Λ generating the next Steel pointclass and yet the Steel pointclass is not closed under disjunction. In the third section, we look at the type IV projective-like hierarchies, that is the pointclass which starts the new hierarchy is closed under quantifiers.