Topologically Invariant Σ-Ideals on the Hilbert Cube
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Certain Ideals Related to the Strong Measure Zero Ideal
数理解析研究所講究録 第 1595 巻 2008 年 37-46 37 Certain ideals related to the strong measure zero ideal 大阪府立大学理学系研究科 大須賀昇 (Noboru Osuga) Graduate school of Science, Osaka Prefecture University 1 Motivation and Basic Definition In 1919, Borel[l] introduced the new class of Lebesgue measure zero sets called strong measure zero sets today. The family of all strong measure zero sets become $\sigma$-ideal and is called the strong measure zero ideal. The four cardinal invariants (the additivity, covering number, uniformity and cofinal- ity) related to the strong measure zero ideal have been studied. In 2002, Yorioka[2] obtained the results about the cofinality of the strong measure zero ideal. In the process, he introduced the ideal $\mathcal{I}_{f}$ for each strictly increas- ing function $f$ on $\omega$ . The ideal $\mathcal{I}_{f}$ relates to the structure of the real line. We are interested in how the cardinal invariants of the ideal $\mathcal{I}_{f}$ behave. $Ma\dot{i}$ly, we te interested in the cardinal invariants of the ideals $\mathcal{I}_{f}$ . In this paper, we deal the consistency problems about the relationship between the cardi- nal invariants of the ideals $\mathcal{I}_{f}$ and the minimam and supremum of cardinal invariants of the ideals $\mathcal{I}_{g}$ for all $g$ . We explain some notation which we use in this paper. Our notation is quite standard. And we refer the reader to [3] and [4] for undefined notation. For sets X and $Y$, we denote by $xY$ the set of all functions $homX$ to Y. -
COVERING PROPERTIES of IDEALS 1. Introduction We Will Discuss The
COVERING PROPERTIES OF IDEALS MAREK BALCERZAK, BARNABAS´ FARKAS, AND SZYMON GLA¸B Abstract. M. Elekes proved that any infinite-fold cover of a σ-finite measure space by a sequence of measurable sets has a subsequence with the same property such that the set of indices of this subsequence has density zero. Applying this theorem he gave a new proof for the random-indestructibility of the density zero ideal. He asked about other variants of this theorem concerning I-almost everywhere infinite-fold covers of Polish spaces where I is a σ-ideal on the space and the set of indices of the required subsequence should be in a fixed ideal J on !. We introduce the notion of the J-covering property of a pair (A;I) where A is a σ- algebra on a set X and I ⊆ P(X) is an ideal. We present some counterexamples, discuss the category case and the Fubini product of the null ideal N and the meager ideal M. We investigate connections between this property and forcing-indestructibility of ideals. We show that the family of all Borel ideals J on ! such that M has the J- covering property consists exactly of non weak Q-ideals. We also study the existence of smallest elements, with respect to Katˇetov-Blass order, in the family of those ideals J on ! such that N or M has the J-covering property. Furthermore, we prove a general result about the cases when the covering property \strongly" fails. 1. Introduction We will discuss the following result due to Elekes [8]. -
Product Specifications and Ordering Information CMS 2019 R2 Condition Monitoring Software
Product specifications and ordering information CMS 2019 R2 Condition Monitoring Software Overview SETPOINT® CMS Condition Monitoring Software provides a powerful, flexible, and comprehensive solution for collection, storage, and visualization of vibration and condition data from VIBROCONTROL 8000 (VC-8000) Machinery Protection System (MPS) racks, allowing trending, diagnostics, and predictive maintenance of monitored machinery. The software can be implemented in three different configurations: 1. CMSPI (PI System) This implementation streams data continuously from all VC-8000 racks into a connected PI Server infrastructure. It consumes on average 12-15 PI tags per connected vibration transducer (refer to 3. manual S1176125). Customers can use their CMSHD/SD (Hard Drive) This implementation is used existing PI ecosystem, or for those without PI, a when a network is not available stand-alone PI server can be deployed as a self- between VC-8000 racks and a contained condition monitoring solution. server, and the embedded flight It provides the most functionality (see Tables 1 recorder in each VC-8000 rack and 2) of the three possible configurations by will be used instead. The flight recorder is a offering full integration with process data, solid-state hard drive local to each VC-8000 rack integrated aero/thermal performance monitoring, that can store anywhere from 1 – 12 months of integrated decision support functionality, nested data*. The data is identical to that which would machine train diagrams and nearly unlimited be streamed to an external server, but remains in visualization capabilities via PI Vision or PI the VC-8000 rack until manually retrieved via ProcessBook, and all of the power of the OSIsoft removable SDHC card media, or by connecting a PI System and its ecosystem of complementary laptop to the rack’s Ethernet CMS port and technologies. -
Contents 3 Homomorphisms, Ideals, and Quotients
Ring Theory (part 3): Homomorphisms, Ideals, and Quotients (by Evan Dummit, 2018, v. 1.01) Contents 3 Homomorphisms, Ideals, and Quotients 1 3.1 Ring Isomorphisms and Homomorphisms . 1 3.1.1 Ring Isomorphisms . 1 3.1.2 Ring Homomorphisms . 4 3.2 Ideals and Quotient Rings . 7 3.2.1 Ideals . 8 3.2.2 Quotient Rings . 9 3.2.3 Homomorphisms and Quotient Rings . 11 3.3 Properties of Ideals . 13 3.3.1 The Isomorphism Theorems . 13 3.3.2 Generation of Ideals . 14 3.3.3 Maximal and Prime Ideals . 17 3.3.4 The Chinese Remainder Theorem . 20 3.4 Rings of Fractions . 21 3 Homomorphisms, Ideals, and Quotients In this chapter, we will examine some more intricate properties of general rings. We begin with a discussion of isomorphisms, which provide a way of identifying two rings whose structures are identical, and then examine the broader class of ring homomorphisms, which are the structure-preserving functions from one ring to another. Next, we study ideals and quotient rings, which provide the most general version of modular arithmetic in a ring, and which are fundamentally connected with ring homomorphisms. We close with a detailed study of the structure of ideals and quotients in commutative rings with 1. 3.1 Ring Isomorphisms and Homomorphisms • We begin our study with a discussion of structure-preserving maps between rings. 3.1.1 Ring Isomorphisms • We have encountered several examples of rings with very similar structures. • For example, consider the two rings R = Z=6Z and S = (Z=2Z) × (Z=3Z). -
Ordinal Numbers and the Well-Ordering Theorem Ken Brown, Cornell University, September 2013
Mathematics 4530 Ordinal numbers and the well-ordering theorem Ken Brown, Cornell University, September 2013 The ordinal numbers form an extension of the natural numbers. Here are the first few of them: 0; 1; 2; : : : ; !; ! + 1;! + 2;:::;! + ! =: !2;!2 + 1;:::;!3;:::;!2;:::: They go on forever. As soon as some initial segment of ordinals has been con- structed, a new one is adjoined that is bigger than all of them. The theory of ordinals is closely related to the theory of well-ordered sets (see Section 10 of Munkres). Recall that a simply ordered set X is said to be well- ordered if every nonempty subset Y has a smallest element, i.e., an element y0 such that y0 ≤ y for all y 2 Y . For example, the following sets are well-ordered: ;; f1g ; f1; 2g ; f1; 2; : : : ; ng ; f1; 2;::: g ; f1; 2;:::;!g ; f1; 2; : : : ; !; ! + 1g : On the other hand, Z is not well-ordered in its natural ordering. As the list of examples suggests, constructing arbitrarily large well-ordered sets is essentially the same as constructing the system of ordinal numbers. Rather than taking the time to develop the theory of ordinals, we will concentrate on well-ordered sets in what follows. Notation. In dealing with ordered sets in what follows we will often use notation such as X<x := fy 2 X j y < xg : A set of the form X<x is called a section of X. Well-orderings are useful because they allow proofs by induction: Induction principle. Let X be a well-ordered set and let Y be a subset. -
D-18 Theˇcech–Stone Compactifications of N and R
d-18 The Cech–Stoneˇ compactifications of N and R 213 d-18 The Cech–Stoneˇ Compactifications of N and R c It is safe to say that among the Cech–Stoneˇ compactifica- The proof that βN has cardinality 2 employs an in- tions of individual spaces, that of the space N of natural dependent family, which we define on the countable set numbers is the most widely studied. A good candidate for {hn, si: n ∈ N, s ⊆ P(n)}. For every subset x of N put second place is βR. This article highlights some of the most Ix = {hn, si: x ∩ n ∈ s}. Now if F and G are finite dis- striking properties of both compactifications. T S joint subsets of P(N) then x∈F Ix \ y∈G Iy is non-empty – this is what independent means. Sending u ∈ βN to the P( ) ∗ point pu ∈ N 2, defined by pu(x) = 1 iff Ix ∈ u, gives us a 1. Description of βN and N c continuous map from βN onto 2. Thus the existence of an independent family of size c easily implies the well-known The space β (aka βω) appeared anonymously in [10] as an N fact that the Cantor cube c2 is separable; conversely, if D is example of a compact Hausdorff space without non-trivial c dense in 2 then setting Iα = {d ∈ D: dα = 0} defines an converging sequences. The construction went as follows: for ∗ ∈ independent family on D. Any closed subset F of N such every x (0, 1) let 0.a1(x)a2(x).. -
Absolute Measurable Spaces ENCYCLOPEDIAOF MATHEMATICSAND ITS APPLICATIONS
A bsolute M easurable S paces Absolutemeasurablespaceandabsolutenull spaceareveryold topological notions, developed from descriptive set theory, topology, Borel measure theory and analysis. This monograph systematically develops and returns to the topological and geometrical origins of these notions. Motivating the development of the exposition aretheaction of thegroup of homeomorphismsof a spaceon Borelmeasures,the Oxtoby–Ulam theorem on Lebesgue-like measures on the unit cube, and the extensions of this theorem to many other topological spaces. Existence of uncountable absolute null space, extension of the Purves theorem, and recent advances on homeomorphic Borel probability measures on the Cantor space are among the many topics discussed. A brief discussion of set-theoretic results on absolutenull spaceis also given. A four-part appendix aids the reader with topological dimension theory, Hausdorff measure and Hausdorff dimension, and geometric measure theory. The exposition will suit researchers and graduate students of real analysis, set theory and measure theory. Togo Nishiurais Professor Emeritus at Wayne State University, Detroit, andAsso- ciate Fell in Mathematics at Dickinson College, Pennsylvania. ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS All the titles listed bel can be obtained from good booksellers or from Cambridge University Press. For a complete series listing visit http://www.cambridge.org/uk/series/sSeries.asp?code=EOM! 63. A. C. Thompson Minkowski Geometry 64 R. B. Bapat and T. E. S. Raghavan Nonnegative Matrices with Applications 65. K. Engel Sperner Theory 66. D. Cvetkovic, P. Rlinson, and S. Simic Eigenspaces of Graphs 67. F. Bergeron, G. Labelle, and P. Leroux Combinational Species and Tree-Like Structures 68. R. Goodman and N. -
Cofinality of the Nonstationary Ideal 0
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 357, Number 12, Pages 4813–4837 S 0002-9947(05)04007-9 Article electronically published on June 29, 2005 COFINALITY OF THE NONSTATIONARY IDEAL PIERRE MATET, ANDRZEJ ROSLANOWSKI, AND SAHARON SHELAH Abstract. We show that the reduced cofinality of the nonstationary ideal NSκ on a regular uncountable cardinal κ may be less than its cofinality, where the reduced cofinality of NSκ is the least cardinality of any family F of nonsta- tionary subsets of κ such that every nonstationary subset of κ can be covered by less than κ many members of F. For this we investigate connections of κ the various cofinalities of NSκ with other cardinal characteristics of κ and we also give a property of forcing notions (called manageability) which is pre- served in <κ–support iterations and which implies that the forcing notion preserves non-meagerness of subsets of κκ (and does not collapse cardinals nor changes cofinalities). 0. Introduction Let κ be a regular uncountable cardinal. For C ⊆ κ and γ ≤ κ,wesaythatγ is a limit point of C if (C ∩ γ)=γ>0. C is closed unbounded if C is a cofinal subset of κ containing all its limit points less than κ.AsetA ⊆ κ is nonstationary if A is disjoint from some closed unbounded subset C of κ. The nonstationary subsets of κ form an ideal on κ denoted by NSκ.Thecofinality of this ideal, cof(NSκ), is the least cardinality of a family F of nonstationary subsets of κ such that every nonstationary subset of κ is contained in a member of F.Thereduced cofinality of NSκ, cof(NSκ), is the least cardinality of a family F⊆NSκ such that every nonstationary subset of κ can be covered by less than κ many members of F.This paper addresses the question of whether cof(NSκ)=cof(NSκ). -
Foundations of Algebraic Geometry Class 6
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 6 RAVI VAKIL CONTENTS 1. More examples of the underlying sets of affine schemes 1 2. The Zariski topology: the underlying topological space of an affine scheme 5 3. Topological definitions 8 Last day: inverse image sheaf; sheaves on a base; toward schemes; the underlying set of an affine scheme. 1. MORE EXAMPLES OF THE UNDERLYING SETS OF AFFINE SCHEMES We are in the midst of discussing the underlying set of an affine scheme. We are looking at examples and learning how to draw pictures. 2 Example 7: AC = Spec C[x; y]. (As with Examples 1 and 2, discussion will apply with C replaced by any algebraically closed field.) Sadly, C[x; y] is not a Principal Ideal Domain: (x; y) is not a principal ideal. We can quickly name some prime ideals. One is (0), which has the same flavor as the (0) ideals in the previous examples. (x - 2; y - 3) is prime, and indeed maximal, because C[x; y]=(x - 2; y - 3) =∼ C, where this isomorphism is via f(x; y) f(2; 3). More generally, (x - a; y - b) is prime for any (a; b) C2. Also, if f(x; y) 7 2 2 3 2 is an irr!educible polynomial (e.g. y - x or y - x ) then (f(x; y)) is prime. 1.A. EXERCISE. (Feel free to skip this exercise, as we will see a different proof of this later.) Show that we have identified all the prime ideals of C[x; y]. We can now attempt to draw a picture of this space. -
Dense Subgroups of Compact Groups
Open Problems in Topology II – Edited by E. Pearl © 2007 Elsevier B.V. All rights reserved Dense subgroups of compact groups W. W. Comfort 1. Introduction The symbol G here denotes the class of all infinite groups, and TG denotes the class of all infinite topological groups which satisfy the T0 separation prop- erty. Each element of TG is, then, a Tychonoff space (i.e., a completely regular, Hausdorff space) [50, 8.4]. We say that G = (G, T ) ∈ TG is totally bounded (some authors prefer the expression pre-compact) if for every U ∈ T \ {∅} there is a finite set F ⊆ G such that G = FU. Our point of departure is the following portion of Weil’s Theorem [77]: Every totally bounded G ∈ TG embeds as a dense topological subgroup of a compact group G; this is unique in the sense that for every compact group Ge containing G densely there is a homeomorphism-and- isomorphism ψ : G Ge fixing G pointwise. As usual, a space is ω-bounded if each of its countable subsets has compact closure. For G = (G, T ) ∈ TG we write G ∈ C [resp., G ∈ Ω; G ∈ CC; G ∈ P; G ∈ TB] if (G, T ) is compact [resp., ω-bounded; countably compact; pseudocompact; totally bounded]. And for X ∈ {C, Ω, CC, P, TB} and G ∈ G we write G ∈ X0 if G admits a group topology T such that (G, T ) ∈ X. The class-theoretic inclusions C ⊆ Ω ⊆ CC ⊆ P ⊆ TB ⊆ TG and C0 ⊆ Ω0 ⊆ CC0 ⊆ P0 ⊆ TB0 ⊆ TG0 = G are easily established. (For P ⊆ TB, see [31]. -
Math Review: Sets, Functions, Permutations, Combinations, and Notation
Math Review: Sets, Functions, Permutations, Combinations, and Notation A.l Introduction A.2 Definitions, Axioms, Theorems, Corollaries, and Lemmas A.3 Elements of Set Theory AA Relations, Point Functions, and Set Functions . A.S Combinations and Permutations A.6 Summation, Integration and Matrix Differentiation Notation A.l Introduction In this appendix we review basic results concerning set theory, relations and functions, combinations and permutations, and summa tion and integration notation. We also review the meaning of the terms defini tion, axiom, theorem, corollary, and lemma, which are labels that are affixed to a myriad of statements and results that constitute the theory of probability and mathematical statistics. The topics reviewed in this appendix constitute basic foundational material on which our study of mathematical statistics will be based. Additional mathematical results, often of a more advanced nature, are introduced throughout the text as the need arises. A.2 Definitions, Axioms, Theorems, Corollaries, and Lemmas The development of the theory of probability and mathematical statistics in volves a considerable number of statements consisting of definitions, axioms, theorems, corollaries, and lemmas. These terms will be used for organizing the various statements and results we will examine into these categories: 1. descriptions of meaning; 2. statements that are acceptable as true without proof; 678 Appendix A Math Review: Sets, Functions, Permutations, Combinations, and Notation 3. formulas or statements that require proof of validity; 4. formulas or statements whose validity follows immediately from other true formulas or statements; and 5. results, generally from other branches of mathematics, whose primary pur pose is to facilitate the proof of validity of formulas or statements in math ematical statistics. -
Introducing Boolean Semilattices Clifford Bergman Iowa State University, [email protected]
Mathematics Publications Mathematics 3-21-2018 Introducing Boolean Semilattices Clifford Bergman Iowa State University, [email protected] Follow this and additional works at: https://lib.dr.iastate.edu/math_pubs Part of the Algebra Commons, and the Logic and Foundations Commons The ompc lete bibliographic information for this item can be found at https://lib.dr.iastate.edu/ math_pubs/195. For information on how to cite this item, please visit http://lib.dr.iastate.edu/ howtocite.html. This Book Chapter is brought to you for free and open access by the Mathematics at Iowa State University Digital Repository. It has been accepted for inclusion in Mathematics Publications by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Introducing Boolean Semilattices Abstract We present and discuss a variety of Boolean algebras with operators that is closely related to the variety generated by all complex algebras of semilattices. We consider the problem of finding a generating set for the variety, representation questions, and axiomatizability. Several interesting subvarieties are presented. We contrast our results with those obtained for a number of other varieties generated by complex algebras of groupoids. Keywords Boolean algebra, BAO, semilattice, Boolean semilattice, Boolean groupoid, canonical extension, equationally definable principal congruence Disciplines Algebra | Logic and Foundations | Mathematics Comments This is a manuscript of a chapter from Bergman C. (2018) Introducing Boolean Semilattices. In: Czelakowski J. (eds) Don Pigozzi on Abstract Algebraic Logic, Universal Algebra, and Computer Science. Outstanding Contributions to Logic, vol 16. Springer, Cham. doi: 10.1007/978-3-319-74772-9_4.