A Note on Baire Spaces and Continuous Lattices
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The Structure of Gunk: Adventures in the Ontology of Space
The Structure of Gunk: Adventures in the Ontology of Space Jeff Sanford Russell February 16, 2009 1 Points and gunk Here are two ways space might be (not the only two): (1) Space is “pointy”. Ev- ery finite region has infinitely many infinitesimal, indivisible parts, called points. Points are zero-dimensional atoms of space. In addition to points, there are other kinds of “thin” boundary regions, like surfaces of spheres. Some regions include their boundaries—the closed regions—others exclude them—the open regions—and others include some bits of boundary and exclude others. Moreover, space includes unextended regions whose size is zero. (2) Space is “gunky”.1 Every region contains still smaller regions—there are no spatial atoms. Every region is “thick”—there are no boundary regions. Every region is extended. Pointy theories of space and space-time—such as Euclidean space or Minkowski space—are the kind that figure in modern physics. A rival tradition, most famously associated in the last century with A. N. Whitehead, instead embraces gunk.2 On the Whiteheadian view, points, curves and surfaces are not parts of space, but rather abstractions from the true regions. Three different motivations push philosophers toward gunky space. The first is that the physical space (or space-time) of our universe might be gunky. We posit spatial reasons to explain what goes on with physical objects; thus the main reason Dean Zimmerman has been a relentless source of suggestions and criticism throughout the writing of this paper. Thanks are also due to Frank Arntzenius, Kenny Easwaran, Peter Forrest, Hilary Greaves, John Hawthorne, and Beatrice Sanford for helpful comments. -
Games in Descriptive Set Theory, Or: It's All Fun and Games Until Someone Loses the Axiom of Choice Hugo Nobrega
Games in Descriptive Set Theory, or: it’s all fun and games until someone loses the axiom of choice Hugo Nobrega Cool Logic 22 May 2015 Descriptive set theory and the Baire space Presentation outline [0] 1 Descriptive set theory and the Baire space Why DST, why NN? The topology of NN and its many flavors 2 Gale-Stewart games and the Axiom of Determinacy 3 Games for classes of functions The classical games The tree game Games for finite Baire classes Descriptive set theory and the Baire space Why DST, why NN? Descriptive set theory The real line R can have some pathologies (in ZFC): for example, not every set of reals is Lebesgue measurable, there may be sets of reals of cardinality strictly between |N| and |R|, etc. Descriptive set theory, the theory of definable sets of real numbers, was developed in part to try to fill in the template “No definable set of reals of complexity c can have pathology P” Descriptive set theory and the Baire space Why DST, why NN? Baire space NN For a lot of questions which interest set theorists, working with R is unnecessarily clumsy. It is often better to work with other (Cauchy-)complete topological spaces of cardinality |R| which have bases of cardinality |N| (a.k.a. Polish spaces), and this is enough (in a technically precise way). The Baire space NN is especially nice, as I hope to show you, and set theorists often (usually?) mean this when they say “real numbers”. Descriptive set theory and the Baire space The topology of NN and its many flavors The topology of NN We consider NN with the product topology of discrete N. -
Topology and Descriptive Set Theory
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector TOPOLOGY AND ITS APPLICATIONS ELSEVIER Topology and its Applications 58 (1994) 195-222 Topology and descriptive set theory Alexander S. Kechris ’ Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA Received 28 March 1994 Abstract This paper consists essentially of the text of a series of four lectures given by the author in the Summer Conference on General Topology and Applications, Amsterdam, August 1994. Instead of attempting to give a general survey of the interrelationships between the two subjects mentioned in the title, which would be an enormous and hopeless task, we chose to illustrate them in a specific context, that of the study of Bore1 actions of Polish groups and Bore1 equivalence relations. This is a rapidly growing area of research of much current interest, which has interesting connections not only with topology and set theory (which are emphasized here), but also to ergodic theory, group representations, operator algebras and logic (particularly model theory and recursion theory). There are four parts, corresponding roughly to each one of the lectures. The first contains a brief review of some fundamental facts from descriptive set theory. In the second we discuss Polish groups, and in the third the basic theory of their Bore1 actions. The last part concentrates on Bore1 equivalence relations. The exposition is essentially self-contained, but proofs, when included at all, are often given in the barest outline. Keywords: Polish spaces; Bore1 sets; Analytic sets; Polish groups; Bore1 actions; Bore1 equivalence relations 1. -
Polish Spaces and Baire Spaces
Polish spaces and Baire spaces Jordan Bell [email protected] Department of Mathematics, University of Toronto June 27, 2014 1 Introduction These notes consist of me working through those parts of the first chapter of Alexander S. Kechris, Classical Descriptive Set Theory, that I think are impor- tant in analysis. Denote by N the set of positive integers. I do not talk about universal spaces like the Cantor space 2N, the Baire space NN, and the Hilbert cube [0; 1]N, or \localization", or about Polish groups. If (X; τ) is a topological space, the Borel σ-algebra of X, denoted by BX , is the smallest σ-algebra of subsets of X that contains τ. BX contains τ, and is closed under complements and countable unions, and rather than talking merely about Borel sets (elements of the Borel σ-algebra), we can be more specific by talking about open sets, closed sets, and sets that are obtained by taking countable unions and complements. Definition 1. An Fσ set is a countable union of closed sets. A Gδ set is a complement of an Fσ set. Equivalently, it is a countable intersection of open sets. If (X; d) is a metric space, the topology induced by the metric d is the topology generated by the collection of open balls. If (X; τ) is a topological space, a metric d on the set X is said to be compatible with τ if τ is the topology induced by d.A metrizable space is a topological space whose topology is induced by some metric, and a completely metrizable space is a topological space whose topology is induced by some complete metric. -
Two Characterizations of Linear Baire Spaces 205
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 45, Number 2, August 1974 TWOCHARACTERIZATIONS OF LINEAR BAIRE SPACES STEPHEN A. SAXON1 ABSTRACT. The Wilansky-Klee conjecture is equivalent to the (un- proved) conjecture that every dense, 1-codimensional subspace of an arbitrary Banach space is a Baire space (second category in itself). The following two characterizations may be useful in dealing with this conjecture: (i) A topological vector space is a Baire space if and on- ly if every absorbing, balanced, closed set is a neighborhood of some point, (ii) A topological vector space is a Baire space if and only if it cannot be covered by countably many nowhere dense sets, each of which is a union of lines (1-dimensional subspaces). Characterization (i) has a more succinct form, using the definition of Wilansky's text [8, p. 224]: a topological vector space is a Baire space if and only if it has the t property. Introduction. The Wilansky-Klee conjecture (see [3], [7]) is equivalent to the conjecture that every dense, 1-codimensional subspace of a Banach space is a Baire space. In [4], [5], [6], [7] it is shown that every counta- ble-codimensional subspace of a locally convex space which is "nearly" a Baire space is, itself, "nearly" a Baire space. (Theorem 1 of this paper indicates how "nearly Baire" a barrelled space is: Wilansky's class of W'barrelled spaces [9,p. 44] is precisely the class of linear Baire spaces.) The theorem [7] that every countable-codimensional subspace of an unor- dered Baire-like space is unordered Baire-like is the closest to an affirmation of the conjecture. -
TOPOLOGY PRELIM REVIEW 2021: LIST THREE Topic 1: Baire Property and Gδ Sets. Definition. X Is a Baire Space If a Countable Inte
TOPOLOGY PRELIM REVIEW 2021: LIST THREE Topic 1: Baire property and Gδ sets. Definition. X is a Baire space if a countable intersection of open, dense subsets of X is dense in X. Complete metric spaces and locally compact spaces are Baire spaces. Def. X is locally compact if for all x 2 X, and all open Ux, there exists Vx with compact closure V x ⊂ Ux. 1. X is locally compact , for all C ⊂ X compact, and all open U ⊃ C, there exists V open with compact closure, so that: C ⊂ V ⊂ V ⊂ U. 2. Def: A set E ⊂ X is nowhere dense in X if its closure E has empty interior. Show: X is a Baire space , any countable union of nowhere dense sets has empty interior. 3. A complete metric space without isolated points is uncountable. (Hint: Baire property, complements of one-point sets.) 4. Uniform boundedness principle. X complete metric, F ⊂ C(X) a family of continuous functions, bounded at each point: (8a 2 X)(9M(a) > 0)(8f 2 F)jf(a)j ≤ M(a): Then there exists a nonempty open set U ⊂ X so that F is eq¨uibounded over U{there exists a constant C > 0 so that: (8f 2 F)(8x 2 U)jf(x)j ≤ C: Hint: For n ≥ 1, consider An = fx 2 X; jf(x)j ≤ n; 8f 2 Fg. Use Baire's property. An important application of 4. is the uniform boundedness principle for families of bounded linear operators F ⊂ L(E; F ), where E; F are Banach spaces. -
Extending Obstructions to Noncommutative Functorial Spectra
EXTENDING OBSTRUCTIONS TO NONCOMMUTATIVE FUNCTORIAL SPECTRA BENNO VAN DEN BERG AND CHRIS HEUNEN Abstract. Any functor from the category of C*-algebras to the category of locales that assigns to each commutative C*-algebra its Gelfand spectrum must be trivial on algebras of n-by-n matrices for n ≥ 3. This obstruction also applies to other spectra such as those named after Zariski, Stone, and Pierce. We extend these no-go results to functors with values in (ringed) topological spaces, (ringed) toposes, schemes, and quantales. The possibility of spectra in other categories is discussed. 1. Introduction The spectrum of a commutative ring is a leading tool of commutative algebra and algebraic geometry. For example, a commutative ring can be reconstructed using (among other ingredients) its Zariski spectrum, a coherent topological space. Spectra are also of central importance to functional analysis and operator algebra. For example, there is a dual equivalence between the category of commutative C*-algebras and compact Hausdorff topological spaces, due to Gelfand.1 A natural question is whether such spectra can be extended to the noncommu- tative setting. Indeed, many candidates have been proposed for noncommutative spectra. In a recent article [23], M. L. Reyes observed that none of the proposed spectra behave functorially, and proved that indeed they cannot, on pain of trivial- izing on the prototypical noncommutative rings Mn(C) of n-by-n matrices with complex entries. To be precise: any functor F : Ringop → Set that satisfies F (C) = Spec(C) for commutative rings C, must also satisfy F (Mn(C)) = ∅ for n ≥ 3. -
On a Notion of Smallness for Subsets of the Baire Space Author(S): Alexander S
On a Notion of Smallness for Subsets of the Baire Space Author(s): Alexander S. Kechris Source: Transactions of the American Mathematical Society, Vol. 229 (May, 1977), pp. 191-207 Published by: American Mathematical Society Stable URL: http://www.jstor.org/stable/1998505 . Accessed: 22/05/2013 14:14 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. American Mathematical Society is collaborating with JSTOR to digitize, preserve and extend access to Transactions of the American Mathematical Society. http://www.jstor.org This content downloaded from 131.215.71.79 on Wed, 22 May 2013 14:14:52 PM All use subject to JSTOR Terms and Conditions TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 229, 1977 ON A NOTION OF SMALLNESS FOR SUBSETS OF THE BAIRE SPACE BY ALEXANDER S. KECHRIS ABSTRACT.Let us call a set A C o' of functionsfrom X into X a-bounded if there is a countablesequence of functions(a.: n E o} C w' such that every memberof A is pointwisedominated by an elementof that sequence. We study in this paper definabilityquestions concerningthis notion of smallnessfor subsets of o'. We show that most of the usual definability results about the structureof countablesubsets of o' have corresponding versionswhich hold about a-boundedsubsets of o'. -
BAIRE SPACES and VIETORIS HYPERSPACES 1. Introduction in This Paper, All Topological Spaces Are Assumed to Be Infinite and at Le
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 135, Number 1, January 2007, Pages 299–303 S 0002-9939(06)08743-0 Article electronically published on August 16, 2006 BAIRE SPACES AND VIETORIS HYPERSPACES JILING CAO, SALVADOR GARC´IA-FERREIRA, AND VALENTIN GUTEV (Communicated by Jonathan M. Borwein) Abstract. We prove that if the Vietoris hyperspace CL(X) of all nonempty closed subsets of a space X is Baire, then all finite powers of X must be Baire spaces. In particular, there exists a metrizable Baire space X whose Vietoris hyperspace CL(X) is not Baire. This settles an open problem of R. A. McCoy stated in 1975. 1. Introduction In this paper, all topological spaces are assumed to be infinite and at least Haus- dorff. Also, all product spaces are endowed with the Tychonoff product topology. A topological space X is called Baire [HM] if the intersection of any sequence of dense open subsets of X is dense in X, or equivalently, if all nonempty open subsets of X are of second category. Baire spaces have numerous applications in analysis and topology, such as the open mapping and closed graph theorems, and the Banach- Steinhaus theorem [Con]. For some other applications, see [E, Za1] or [Za2]. A fundamental treatise on Baire spaces in general topology is [HM], and several open problems on Baire and related spaces are discussed in [AL]. For a space X,letCL(X) be the collection of all nonempty closed subsets of X endowed with the Vietoris topology τV . Recall that a canonical base for τV is given by all subsets of CL(X)oftheform V = F ∈ CL(X):F ⊂ V,F∩ V = ∅ for any V ∈V , where V runs over the finite families of nonempty open subsets of X.Inthesequel, any subset D⊂CL(X) will carry the relative Vietoris topology τV as a subspace of (CL(X),τV ). -
Regularity Properties and Determinacy
Regularity Properties and Determinacy MSc Thesis (Afstudeerscriptie) written by Yurii Khomskii (born September 5, 1980 in Moscow, Russia) under the supervision of Dr. Benedikt L¨owe, and submitted to the Board of Examiners in partial fulfillment of the requirements for the degree of MSc in Logic at the Universiteit van Amsterdam. Date of the public defense: Members of the Thesis Committee: August 14, 2007 Dr. Benedikt L¨owe Prof. Dr. Jouko V¨a¨an¨anen Prof. Dr. Joel David Hamkins Prof. Dr. Peter van Emde Boas Brian Semmes i Contents 0. Introduction............................ 1 1. Preliminaries ........................... 4 1.1 Notation. ........................... 4 1.2 The Real Numbers. ...................... 5 1.3 Trees. ............................. 6 1.4 The Forcing Notions. ..................... 7 2. ClasswiseConsequencesofDeterminacy . 11 2.1 Regularity Properties. .................... 11 2.2 Infinite Games. ........................ 14 2.3 Classwise Implications. .................... 16 3. The Marczewski-Burstin Algebra and the Baire Property . 20 3.1 MB and BP. ......................... 20 3.2 Fusion Sequences. ...................... 23 3.3 Counter-examples. ...................... 26 4. DeterminacyandtheBaireProperty.. 29 4.1 Generalized MB-algebras. .................. 29 4.2 Determinacy and BP(P). ................... 31 4.3 Determinacy and wBP(P). .................. 34 5. Determinacy andAsymmetric Properties. 39 5.1 The Asymmetric Properties. ................. 39 5.2 The General Definition of Asym(P). ............. 43 5.3 Determinacy and Asym(P). ................. 46 ii iii 0. Introduction One of the most intriguing developments of modern set theory is the investi- gation of two-player infinite games of perfect information. Of course, it is clear that applied game theory, as any other branch of mathematics, can be modeled in set theory. But we are talking about the converse: the use of infinite games as a tool to study fundamental set theoretic questions. -
[Math.GN] 4 Sep 2006 Aiod,Toewihaehudr.W Hwta Hsi O H Ca the Not the Is Called This That first, Show We the Hausdorff
MANIFOLDS: HAUSDORFFNESS VERSUS HOMOGENEITY MATHIEU BAILLIF AND ALEXANDRE GABARD Abstract. We analyze the relationship between Hausdorffness and homogeneity in the frame of manifolds, not confined to be Hausdorff. We exhibit examples of homo- geneous non-Hausdorff manifolds and prove that a Lindel¨of homogeneous manifold is Hausdorff. 2000 Mathematics Subject Classification. 57N99, 54D10, 54E52. Key words and phrases. Manifolds, Non-Hausdorff manifolds, homogeneity. 1 Introduction Our purpose here is to analyze the relationship between Hausdorffness and homogeneity in the frame of manifolds. We give the word manifold its broadest sense, that is, a topological space locally homeomorphic to the Euclidean space Rn of a fixed dimension (without assuming the Hausdorff separation axiom). Recall that a connected Hausdorff manifold M is homogeneous, i.e. for each x, y ∈ M, there is a homeomorphism h : M → M taking x to y (see [8] or [9], p. 150). This property is true only under the Hausdorff assumption. Without it, one may well have an non homogeneous manifold, for example the well known line with two origins: take two copies of the real line R and identify all corresponding points of the copies but the origin (Figure 1). This yields a one-dimensional manifold in which the two origins cannot be separated1. Notice though that a point different from the origins can be separated from any other point, so the manifold is not homogeneous. Another well known example of non-Hausdorff manifold is the branching line obtained by identifying the points < 0 in the two copies of R (Figure 1). no identifications ... ... ... ... no identifications .. -
Baire Spaces and Graph Theorems
PROCEEDINGS of the AMERICAN MATHEMATICAL SOCIETY Volume 96. Number 1, January 1986 BAIRE SPACES AND GRAPH THEOREMS DOMINIK NOLL Abstract. We prove graph theorems and a variant of the Banach-Steinhaus theorem in a purely topological context. We obtain characterizations of Baire spaces in the class of metrizable spaces by means of graph theorems. Introduction. The following abstract form of a graph theorem covers the pre- ponderant part of work in this field: Let f be a nearly continuous mapping from a space E to a space F which satisfies a certain condition (frequently expressed in terms of the graph of f). Then, under appropriate conditions on E and F, the mapping f is continuous. The most convenient applications of this scheme deal with the case where the graph of the mapping / is closed. The reader might consult [Ko, p. 33ff], where the classical theory is treated, [Hu] for the situation in topological groups, [Ke], [LR] for the case of uniform spaces, [Wi], [BP], [We] for the general topological case. In this paper, the main interest is directed toward the case where the graph G(f) of/is a Gs-set (more generally a set of interior condensation). In §1 we prove that if Tí is a Baire space and if F is complete (in a certain sense) then the graph theorem above holds for mappings / whose graph is Gs. In §2 we prove graph theorems for mappings with measurability conditions. In this case the completeness property for F is no longer needed. As a consequence, in §2 we obtain a purely topological variant of the Banach-Steinhaus theorem.