Some Operators on Uniform Spaces
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A Guide to Topology
i i “topguide” — 2010/12/8 — 17:36 — page i — #1 i i A Guide to Topology i i i i i i “topguide” — 2011/2/15 — 16:42 — page ii — #2 i i c 2009 by The Mathematical Association of America (Incorporated) Library of Congress Catalog Card Number 2009929077 Print Edition ISBN 978-0-88385-346-7 Electronic Edition ISBN 978-0-88385-917-9 Printed in the United States of America Current Printing (last digit): 10987654321 i i i i i i “topguide” — 2010/12/8 — 17:36 — page iii — #3 i i The Dolciani Mathematical Expositions NUMBER FORTY MAA Guides # 4 A Guide to Topology Steven G. Krantz Washington University, St. Louis ® Published and Distributed by The Mathematical Association of America i i i i i i “topguide” — 2010/12/8 — 17:36 — page iv — #4 i i DOLCIANI MATHEMATICAL EXPOSITIONS Committee on Books Paul Zorn, Chair Dolciani Mathematical Expositions Editorial Board Underwood Dudley, Editor Jeremy S. Case Rosalie A. Dance Tevian Dray Patricia B. Humphrey Virginia E. Knight Mark A. Peterson Jonathan Rogness Thomas Q. Sibley Joe Alyn Stickles i i i i i i “topguide” — 2010/12/8 — 17:36 — page v — #5 i i The DOLCIANI MATHEMATICAL EXPOSITIONS series of the Mathematical Association of America was established through a generous gift to the Association from Mary P. Dolciani, Professor of Mathematics at Hunter College of the City Uni- versity of New York. In making the gift, Professor Dolciani, herself an exceptionally talented and successfulexpositor of mathematics, had the purpose of furthering the ideal of excellence in mathematical exposition. -
On Finite-Dimensional Uniform Spaces
Pacific Journal of Mathematics ON FINITE-DIMENSIONAL UNIFORM SPACES JOHN ROLFE ISBELL Vol. 9, No. 1 May 1959 ON FINITE-DIMENSIONAL UNIFORM SPACES J. R. ISBELL Introduction* This paper has two nearly independent parts, con- cerned respectively with extension of mappings and with dimension in uniform spaces. It is already known that the basic extension theorems of point set topology are valid in part, and only in part, for uniformly continuous functions. The principal contribution added here is an affirmative result to the effect that every finite-dimensional simplicial complex is a uniform ANR, or ANRU. The complex is supposed to carry the uniformity in which a mapping into it is uniformly continuous if and only if its barycentric coordinates are equiuniformly continuous. (This is a metric uniformity.) The conclusion (ANRU) means that when- ever this space μA is embedded in another uniform space μX there exist a uniform neighborhood U of A (an ε-neighborhood with respect to some uniformly continuous pseudometric) and a uniformly continuous retrac- tion r : μU -> μA, It is known that the real line is not an ARU. (Definition obvious.) Our principal negative contribution here is the proof that no uniform space homeomorphic with the line is an ARU. This is also an indication of the power of the methods, another indication being provided by the failure to settle the corresponding question for the plane. An ARU has to be uniformly contractible, but it does not have to be uniformly locally an ANRU. (The counter-example is compact metric and is due to Borsuk [2]). -
Strong Shape of Uniform Spaces
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 49 (1993) 237-249 237 North-Holland Strong shape of uniform spaces Jack Segal Department of Mathematics, University of Washington, Cl38 Padeljord Hall GN-50, Seattle, WA 98195, USA Stanislaw Spiei” Institute of Mathematics, PAN, Warsaw, Poland Bernd Giinther”” Fachbereich Mathematik, Johann Wolfgang Goethe-Universitiit, Robert-Mayer-Strafie 6-10, W-6000 Frankfurt, Germany Received 19 August 1991 Revised 16 March 1992 Abstract Segal, J., S. Spiei and B. Giinther, Strong shape of uniform spaces, Topology and its Applications 49 (1993) 237-249. A strong shape category for finitistic uniform spaces is constructed and it is shown, that certain nice properties known from strong shape theory of compact Hausdorff spaces carry over to this setting. These properties include a characterization of the new category as localization of the homotopy category, the product property and obstruction theory. Keywords: Uniform spaces, finitistic uniform spaces, strong shape, Cartesian products, localiz- ation, Samuel compactification, obstruction theory. AMS (MOS) Subj. Class.: 54C56, 54835, 55N05, 55P55, 55S35. Introduction In strong shape theory of topological spaces one encounters several instances, where the desired extension of theorems from compact spaces to more general ones either leads to difficult unsolved problems or is outright impossible. Examples are: Correspondence to: Professor J. Segal, Department of Mathematics, University of Washington, Seattle, WA 98195, USA. * This paper was written while this author was visiting the University of Washington. ** This paper was written while this author was visiting the University of Washington, and this visit was supported by a DFG fellowship. -
Notes on Uniform Structures Annex to H104
Notes on Uniform Structures Annex to H104 Mariusz Wodzicki December 13, 2013 1 Vocabulary 1.1 Binary Relations 1.1.1 The power set Given a set X, we denote the set of all subsets of X by P(X) and by P∗(X) — the set of all nonempty subsets. The set of subsets E ⊆ X which contain a given subset A will be denoted PA(X). If A , Æ, then PA(X) is a filter. Note that one has PÆ(X) = P(X). 1.1.2 In these notes we identify binary relations between elements of a set X and a set Y with subsets E ⊆ X×Y of their Cartesian product X×Y. To a given relation ∼ corresponds the subset: E∼ ˜ f(x, y) 2 X×Y j x ∼ yg (1) and, vice-versa, to a given subset E ⊆ X×Y corresponds the relation: x ∼E y if and only if (x, y) 2 E.(2) 1.1.3 The opposite relation We denote by Eop ˜ f(y, x) 2 Y×X j (x, y) 2 Eg (3) the opposite relation. 1 1.1.4 The correspondence E 7−! Eop (E ⊆ X×X) (4) defines an involution1 of P(X×X). It induces the corresponding involu- tion of P(P(X×X)): op E 7−! op∗(E ) ˜ fE j E 2 E g (E ⊆ P(X×X)).(5) Exercise 1 Show that op∗(E ) (a) possesses the Finite Intersection Property, if E possesses the Finite Intersec- tion Property; (b) is a filter-base, if E is a filter-base; (c) is a filter, if E is a filter. -
Vanishing Cycles for Formal Schemes
Invent math. 115, 539 571 (1994) Inventiones mathematicae Springer-Verlag 1994 Vanishing cycles for formal schemes Vladimir G. Berkovich * Department of Theoretical Mathematics, The WeizmannInstitute of Science, P.O.B. 26, 76100 Rehovot, Israel Oblatum 4-VI-1993 & 6-VIli-1993 Introduction Let k be a non-Archimedean field, and let X be a formal scheme locally finitely presented over the ring of integers k ~ (see w In this work we construct and study the vanishing cycles functor from the category of 6tale sheaves on the generic fibre X~ of ff (which is a k-analytic space) to the category of 6tale sheaves on the closed fibre 3~ of Y (which is a scheme over the residue field of k). We prove that if ff is the formal completion ~' of a scheme X finitely presented over k ~ along the closed fibre, then the vanishing cycles sheaves of ~" are canonically isomorphic to those of P( (as defined in [SGA7], Exp. XIII). In particular, the vanishing cycles sheaves of A" depend only on ,~, and any morphism ~ : ~ -~ A' induces a homomorphism from the pullback of the vanishing cycles sheaves of A2 under ~ : y~ --~ X~ to those of Y. Furthermore, we prove that, for each ~', one can find a nontrivial ideal of k ~ such that if two morphisms ~, ~ : ~ ~ ,~ coincide modulo this ideal, then the homomorphisms between the vanishing cycles sheaves induced by ~ and g) coincide. These facts were conjectured by P. Deligne. In w we associate with a formal scheme Y locally finitely presented over k ~ a k-analytic space X,j (in the sense of [Berl] and [Ber2]). -
A Uniformities and Uniform Completion
A Uniformities and Uniform Completion A uniform space is a set X together with a structure called a uniformity defined on it. This structure can be defined in three different but equivalent ways, each with its own particular advantages.1 The first two of the following originate in the seminal work of Weil [109]; the third was defined by Tukey [108].2 1. In terms of a family of pseudometrics on X. This definition is used by Gillman and Jerison in their monograph on Rings of Continuous Functions [43], “as they [pseudometrics] provide us with a large supply of continuous functions”. 2. As a filter on X × X that satisfies certain conditions, one of them being that each member of the filter contain the diagonal Δ = {(x, x)|x ∈ X}. This definition was given in its present form by Bourbaki [12]. 3. In terms of certain covers of X called uniform covers,inwhichX is covered by “sets of the same size”. This definition was given by Tukey [108], and is used extensively by Isbell in his monograph [53] on what, according to its author, “might be labelled fairly accurately [as the] intrinsic geometry of uniform spaces”. All three definitions are given below. It should be pointed out that there are slight differences in terminology in the literature, the most important 1 Willard [130] calls the second and third of these diagonal and covering unifor- mities, respectively, and we shall follow his terminology. In the same spirit, we shall call the first pseudometric uniformities. It should be borne in mind that the different names refer, not to different mathematical objects, but to different definitions of the same object. -
Expansive Actions with Specification on Uniform Spaces, Topological Entropy, and the Myhill Property
EXPANSIVE ACTIONS WITH SPECIFICATION ON UNIFORM SPACES, TOPOLOGICAL ENTROPY, AND THE MYHILL PROPERTY TULLIO CECCHERINI-SILBERSTEIN AND MICHEL COORNAERT Abstract. We prove that every expansive continuous action with the weak specification property of an amenable group G on a compact Hausdorff space X has the Myhill property, i.e., every pre-injective continuous self-mapping of X commuting with the action of G on X is surjective. This extends a result previously obtained by Hanfeng Li in the case when X is metrizable. Contents 1. Introduction 2 2. Background material 4 2.1. General notation 4 2.2. Subsets of X × X 4 2.3. Uniform spaces 5 2.4. Ultrauniform spaces 6 2.5. Actions 7 2.6. Covers 7 2.7. Shift spaces 8 2.8. Amenable groups 8 3. Expansiveness 9 4. Topological entropy 10 4.1. Topological entropy 10 4.2. (F, U)-separated subsets, (F, U)-spanning subsets, and (F, U)-covers 12 4.3. Uniformapproachestotopologicalentropy 14 4.4. Topological entropy and expansiveness 17 5. Homoclinicity 19 6. Weak specification 20 arXiv:1905.02740v3 [math.DS] 6 Mar 2020 7. Proof of the main result 25 References 31 Date: March 9, 2020. 2010 Mathematics Subject Classification. 54E15, 37B05, 37B10, 37B15, 37B40, 43A07, 68Q80. Key words and phrases. uniform space, uniformly continuous action, expansiveness, specification, amenable group, entropy, homoclinicity, strongly irreducible subshift, cellular automaton. 1 2 TULLIO CECCHERINI-SILBERSTEIN AND MICHEL COORNAERT 1. Introduction A topological dynamical system is a pair (X,G), where X is a topological space and G is a group acting continuously on X. -
Topology Proceedings 9 (1984) Pp. 61-83: on UNIFORM HYPERSPACES
Volume 9, 1984 Pages 61{83 http://topology.auburn.edu/tp/ ON UNIFORM HYPERSPACES by Aarno Hohti Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: [email protected] ISSN: 0146-4124 COPYRIGHT °c by Topology Proceedings. All rights reserved. TOPOLOGY PROCEEDINGS Volume 9 1984 61 ON UNIFORM HYPERSPACES Aarno Hohti1 1. Introduction This paper is a continuation of the work done during the 1970's by Z. Frolik, A. Hager, M. Husek, J. Pelant, M. Rice and others on uniform spaces. While the topological study of hyperspaces is a thriving part of mathematics, in the field of uniform topology the uniform hyperspaces have been left essentially unstudied except for John Isbell's characterization of uniform spaces whose uniform hyperspace of all nonempty closed subsets is a complete uniform space, see [7]. Our aim is to fill a part of this gap by consider ing questions related to recent research on uniform spaces. In addition to other results, we characterize the class of uniform spaces whose hyperspaces are metric-fine and the result follows from the fact that the metric-completion of Morita and Rice commutes with the operation of forming the uniform hyperspace of all nonempty compact subsets. 2. Some Preliminary Definitions The reader may consult [8] for information on uniform spaces. A set X with a uniformity ~ is called a uniform space and denoted by ~X. In this paper all uniform spaces louring the preparation of this paper, the author visited the Czechoslovak Academy of Sciences under a scien tific exchange program between Czechoslovakia and Finland, and wishes to thank CSAV, the Academy of Finland and the Finnish Academy of Sciences for support. -
Fundamental Groups and Finite Sheeted Coverings$
Fundamental groups and finite sheeted coveringsI L. Javier Hern´andezParicioa, Vlasta Matijevi´cb aDepartment of Mathematics and Computer Science, University of La Rioja, c/Luis de Ulloa s/n, 26004 Logro~no.Spain bDepartment of Mathematics, University of Split, N. Tesle 12, 21000 Split, Croatia Abstract It is well known that for a connected locally path-connected semilocally 1- connected space X, there exists a bi-unique correspondence between the pointed d-fold connected coverings and the transitive representations of the fundamental group of X in the symmetric group Σd of degree d . The classification problem becomes more difficult if X is a more general space, particularly if X is not locally connected. In attempt to solve the prob- lem for general spaces, several notions of coverings have been introduced, for example, those given by Lubkin or by Fox. On the other hand, different no- tions of `fundamental group' have appeared in the mathematical literature, for instance, the Brown-Grossman-Quigley fundamental group, the Cech-Borsukˇ fundamental group, the Steenrod-Quigley fundamental group, the fundamental profinite group or the fundamental localic group. The main result of this paper determines different `fundamental groups' that can be used to classify pointed finite sheeted connected coverings of a given space X depending on topological properties of X: Key words: Brown-Grossman group, Steenrod group, fundamental progrups, overlay, covering projection 2000 MSC: 55P57, 55Q52, 55Q70, 55P55, 55U40, 55Q07, 55R65 Introduction If X is a connected locally path-connected and semilocally 1-connected space 0 0 0 and π1(X; x ); x 2 X; its fundamental group, then the category Cov (X; x ) of coverings and continuous maps over (X; x0) is equivalent to the category 0 0 π1(X;x ) 0 of π1(X; x )-sets, i.e. -
Coarse Cohomology with Twisted Coefficients Elisa Hartmann
Coarse Cohomology with twisted Coefficients Dissertation zur Erlangung des mathematisch-naturwissenschaftlichen Doktorgrades "Doctor rerum naturalium" der Georg-August-Universität Göttingen im Promotionsprogramm Mathematical Science der Georg-August University School of Science (GAUSS) vorgelegt von Elisa Hartmann aus Greifswald Göttingen, Dezember 2018 Betreuungsausschuss Professor Dr. Laurent Bartholdi, Mathematisches Institut, Georg-August-Universität Göttingen Professor Dr. Ingo Witt, Mathematisches Institut, Georg-August-Universität Göttingen Mitglieder der Prüfungskomission Referent: Professor Dr. Laurent Bartholdi Korreferent: Professor Dr. Ingo Witt Weitere Mitglieder der Prüfungskomission Professor Dr. Max Wardetzky, Institut für Numerische und Angewandte Mathematik, Georg-August-Universität Göttingen Professor Dr. Madeleine Jotz Lean, Mathematisches Institut, Georg-August-Universität Göttingen Professor Dr. Gerlind Plonka-Hoch, Institut für Numerische und Angewandte Mathematik, Georg-August-Universität Göttingen Dr. Christopher Wulff, Mathematisches Institut, Georg-August-Universität Göttingen Tag der mündlichen Prüfung: 25.02.2019 2 Contents 1 Introduction 5 1.1 What is Coarse Geometry? . 5 1.2 Background and related Theories . 6 1.2.1 Cohomology theories . 6 1.2.2 Boundaries . 8 1.3 Main Contributions . 10 1.3.1 Sheaf Cohomology on Coarse Spaces . 10 1.3.2 Space at Infinity . 12 2 Coarse Cohomology with twisted Coefficients 15 2.1 The Coarse Category . 15 2.1.1 Coarse Spaces . 15 2.1.2 Coarse Maps . 17 2.2 Coentourages . 18 2.2.1 Definition . 18 2.2.2 A Discussion/ Useful to know . 19 2.2.3 On Maps . 21 2.3 Limits and Colimits . 24 2.3.1 The Forgetful Functor . 25 2.3.2 Limits . 25 2.3.3 Colimits . 26 2.4 Coarse Cohomology with twisted Coefficients . -
Pyknotic Objects, I. Basic Notions
Pyknotic objects, I. Basic notions Clark Barwick Peter Haine April 30, 2019 Abstract Pyknotic objects are (hyper)sheaves on the site of compacta. These provide a convenient way to do algebra and homotopy theory with additional topological in- formation present. This appears, for example, when trying to contemplate the de- rived category of a local field. In this article, we present the basic theory of pyknotic objects, with a view to describing a simple set of everyday examples. Contents 0 Introduction 2 0.1 The proétale topology and pyknotic objects ................ 2 0.2 The aims of this paper ........................... 3 0.3 Pyknotic and condensed .......................... 4 0.4 Acknowledgements ............................. 4 1 Conventions 5 1.1 Higher categories .............................. 5 1.2 Set theoretic conventions ......................... 5 1.3 Sites and sheaves .............................. 6 1.4 Accessible sheaves ............................. 8 2 Pyknotic objects 10 2.1 Pyknotic sets ................................ 10 2.2 Pyknotic spaces ............................... 13 2.3 Pyknotic objects .............................. 17 2.4 Pyknotic objects of topoi .......................... 20 2.5 Tensor products of pyknotic objects .................... 22 2.6 Pyk-modules ................................ 23 3 Pyknotic objects in algebra & homotopy theory 25 3.1 Pyknotic spectra & pyknotic homotopy groups .............. 25 3.2 Pyknotic rings and pyknotic modules ................... 27 3.3 The proétale topos as a Pyk-algebra .................... 28 1 4 Pyknotic categories 31 4.1 Pyknotic categories ............................. 31 4.2 Pyknotic categories and complete Segal pyknotic spaces ......... 33 4.3 Ultracategories as pseudopyknotic categories ............... 34 References 37 0 Introduction 0.1 The proétale topology and pyknotic objects Let 퐸 be a local field, and let 푋 be a connected, topologically noetherian, coherent scheme. -
Arxiv:1706.00279V1 [Math.GN]
THE SAMUEL REALCOMPACTIFICATION M. ISABEL GARRIDO AND ANA S. MERONO˜ Abstract. For a uniform space (X, µ), we introduce a realcompactification of X by means of the family Uµ(X) of all the real-valued uniformly continuous functions, in the same way that the known Samuel com- ∗ pactification is given by Uµ(X) the set of all the bounded functions in Uµ(X). We will call it “the Samuel realcompactification” by several resemblances to the Samuel compactification. In this note, we present dif- ferent ways to construct such realcompactification as well as we study the corresponding problem of knowing when a uniform space is Samuel realcompact, that is, it coincides with its Samuel realcompactification. At this respect we obtain as main result a theorem of Katˇetov-Shirota type, by means of a new property of completeness recently introduced by the authors, called Bourbaki-completeness. 1. Introduction A realcompactification of a Tychonoff space X is a realcompact space Y in which X is densely embedded. For instance, the well-known Hewitt-Nachbin realcompactification υX. Recall that υX is characterized as the smallest realcompactification of X (in the usual order on the family of all the realcompactifications) such that every real-valued continuous function f ∈ C(X) can be continuously extended to it [10]. In the frame of uniform spaces, since we can also consider Uµ(X), the set of all the real-valued uniformly continuous functions on the uniform space (X,µ), it is natural to ask what is the smallest realcompactification of X such that every function f ∈ Uµ(X) can be continuously extended to it.