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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. -
Quantum Heaps, Cops and Heapy Categories
Mathematical Communications 12(2007), 1-9 1 Quantum heaps, cops and heapy categories Zoran Skodaˇ ∗ Abstract. A heap is a structure with a ternary operation which is intuitively a group with a forgotten unit element. Quantum heaps are associative algebras with a ternary cooperation which are to Hopf algebras what heaps are to groups, and, in particular, the category of copointed quantum heaps is isomorphic to the category of Hopf algebras. There is an intermediate structure of a cop in a monoidal category which is in the case of vector spaces to a quantum heap about what a coalge- bra is to a Hopf algebra. The representations of Hopf algebras make a rigid monoidal category. Similarly, the representations of quantum heaps make a kind of category with ternary products, which we call a heapy category. Key words: quantum algebra, rings, algebras AMS subject classifications: 20N10, 16A24 Received January 25, 2007 Accepted February 1, 2007 1. Classical background: heaps A reference for this section is [1]. 1.1 Before forgetting: group as heap. Let G be a group. Then the ternary operation t : G × G × G → G given by t(a, b, c)=ab−1c, (1) satisfies the following relations: t(b, b, c)=c = t(c, b, b) (2) t(a, b, t(c, d, e)) = t(t(a, b, c),d,e) Aheap(H, t) is a pair of nonempty set H and a ternary operation t : H × H × H satisfying relation (2). A morphism f :(H, t) → (H,t) of heaps is a set map f : H → H satisfying t ◦ (f × f × f)=f ◦ t. -
On Dimension and Weight of a Local Contact Algebra
Filomat 32:15 (2018), 5481–5500 Published by Faculty of Sciences and Mathematics, https://doi.org/10.2298/FIL1815481D University of Nis,ˇ Serbia Available at: http://www.pmf.ni.ac.rs/filomat On Dimension and Weight of a Local Contact Algebra G. Dimova, E. Ivanova-Dimovaa, I. D ¨untschb aFaculty of Math. and Informatics, Sofia University, 5 J. Bourchier Blvd., 1164 Sofia, Bulgaria bCollege of Maths. and Informatics, Fujian Normal University, Fuzhou, China. Permanent address: Dept. of Computer Science, Brock University, St. Catharines, Canada Abstract. As proved in [16], there exists a duality Λt between the category HLC of locally compact Hausdorff spaces and continuous maps, and the category DHLC of complete local contact algebras and appropriate morphisms between them. In this paper, we introduce the notions of weight wa and of dimension dima of a local contact algebra, and we prove that if X is a locally compact Hausdorff space then t t w(X) = wa(Λ (X)), and if, in addition, X is normal, then dim(X) = dima(Λ (X)). 1. Introduction According to Stone’s famous duality theorem [43], the Boolean algebra CO(X) of all clopen (= closed and open) subsets of a zero-dimensional compact Hausdorff space X carries the whole information about the space X, i.e. the space X can be reconstructed from CO(X), up to homeomorphism. It is natural to ask whether the Boolean algebra RC(X) of all regular closed subsets of a compact Hausdorff space X carries the full information about the space X (see Example 2.5 below for RC(X)). -
Ordered Groupoids and the Holomorph of an Inverse Semigroup
ORDERED GROUPOIDS AND THE HOLOMORPH OF AN INVERSE SEMIGROUP N.D. GILBERT AND E.A. MCDOUGALL ABSTRACT. We present a construction for the holomorph of an inverse semi- group, derived from the cartesian closed structure of the category of ordered groupoids. We compare the holomorph with the monoid of mappings that pre- serve the ternary heap operation on an inverse semigroup: for groups these two constructions coincide. We present detailed calculations for semilattices of groups and for the polycyclic monoids. INTRODUCTION The holomorph of a group G is the semidirect product of Hol(G) = Aut(G) n G of G and its automorphism group (with the natural action of Aut(G) on G). The embedding of Aut(G) into the symmetric group ΣG on G extends to an embedding of Hol(G) into ΣG where g 2 G is identified with its (right) Cayley representation ρg : a 7! ag. Then Hol(G) is the normalizer of G in ΣG ([15, Theorem 9.17]). A second interesting characterization of Hol(G) is due to Baer [1] (see also [3] and [15, Exercise 520]). The heap operation on G is the ternary operation defined by ha; b; ci = ab−1c. Baer shows that a subset of G is closed under the heap operation if and only if it is a coset of some subgroup, and that the subset of ΣG that preserves h· · ·i is precisely Hol(G): that is, if σ 2 ΣG, then for all a; b; c; 2 G we have ha; b; ciσ = haσ; bσ; cσi if and only if σ 2 Hol(G). -
Second-Order Algebraic Theories (Extended Abstract)
Second-Order Algebraic Theories (Extended Abstract) Marcelo Fiore and Ola Mahmoud University of Cambridge, Computer Laboratory Abstract. Fiore and Hur [10] recently introduced a conservative exten- sion of universal algebra and equational logic from first to second order. Second-order universal algebra and second-order equational logic respec- tively provide a model theory and a formal deductive system for lan- guages with variable binding and parameterised metavariables. This work completes the foundations of the subject from the viewpoint of categori- cal algebra. Specifically, the paper introduces the notion of second-order algebraic theory and develops its basic theory. Two categorical equiva- lences are established: at the syntactic level, that of second-order equa- tional presentations and second-order algebraic theories; at the semantic level, that of second-order algebras and second-order functorial models. Our development includes a mathematical definition of syntactic trans- lation between second-order equational presentations. This gives the first formalisation of notions such as encodings and transforms in the context of languages with variable binding. 1 Introduction Algebra started with the study of a few sample algebraic structures: groups, rings, lattices, etc. Based on these, Birkhoff [3] laid out the foundations of a general unifying theory, now known as universal algebra. Birkhoff's formalisation of the notion of algebra starts with the introduction of equational presentations. These constitute the syntactic foundations of the subject. Algebras are then the semantics or model theory, and play a crucial role in establishing the logical foundations. Indeed, Birkhoff introduced equational logic as a sound and complete formal deductive system for reasoning about algebraic structure. -
MTH 304: General Topology Semester 2, 2017-2018
MTH 304: General Topology Semester 2, 2017-2018 Dr. Prahlad Vaidyanathan Contents I. Continuous Functions3 1. First Definitions................................3 2. Open Sets...................................4 3. Continuity by Open Sets...........................6 II. Topological Spaces8 1. Definition and Examples...........................8 2. Metric Spaces................................. 11 3. Basis for a topology.............................. 16 4. The Product Topology on X × Y ...................... 18 Q 5. The Product Topology on Xα ....................... 20 6. Closed Sets.................................. 22 7. Continuous Functions............................. 27 8. The Quotient Topology............................ 30 III.Properties of Topological Spaces 36 1. The Hausdorff property............................ 36 2. Connectedness................................. 37 3. Path Connectedness............................. 41 4. Local Connectedness............................. 44 5. Compactness................................. 46 6. Compact Subsets of Rn ............................ 50 7. Continuous Functions on Compact Sets................... 52 8. Compactness in Metric Spaces........................ 56 9. Local Compactness.............................. 59 IV.Separation Axioms 62 1. Regular Spaces................................ 62 2. Normal Spaces................................ 64 3. Tietze's extension Theorem......................... 67 4. Urysohn Metrization Theorem........................ 71 5. Imbedding of Manifolds.......................... -
Chapter 7 Separation Properties
Chapter VII Separation Axioms 1. Introduction “Separation” refers here to whether or not objects like points or disjoint closed sets can be enclosed in disjoint open sets; “separation properties” have nothing to do with the idea of “separated sets” that appeared in our discussion of connectedness in Chapter 5 in spite of the similarity of terminology.. We have already met some simple separation properties of spaces: the XßX!"and X # (Hausdorff) properties. In this chapter, we look at these and others in more depth. As “more separation” is added to spaces, they generally become nicer and nicer especially when “separation” is combined with other properties. For example, we will see that “enough separation” and “a nice base” guarantees that a space is metrizable. “Separation axioms” translates the German term Trennungsaxiome used in the older literature. Therefore the standard separation axioms were historically named XXXX!"#$, , , , and X %, each stronger than its predecessors in the list. Once these were common terminology, another separation axiom was discovered to be useful and “interpolated” into the list: XÞ"" It turns out that the X spaces (also called $$## Tychonoff spaces) are an extremely well-behaved class of spaces with some very nice properties. 2. The Basics Definition 2.1 A topological space \ is called a 1) X! space if, whenever BÁC−\, there either exists an open set Y with B−Y, CÂY or there exists an open set ZC−ZBÂZwith , 2) X" space if, whenever BÁC−\, there exists an open set Ywith B−YßCÂZ and there exists an open set ZBÂYßC−Zwith 3) XBÁC−\Y# space (or, Hausdorff space) if, whenever , there exist disjoint open sets and Z\ in such that B−YC−Z and . -
Foundations of Algebraic Theories and Higher Dimensional Categories
Foundations of Algebraic Theories and Higher Dimensional Categories Soichiro Fujii arXiv:1903.07030v1 [math.CT] 17 Mar 2019 A Doctor Thesis Submitted to the Graduate School of the University of Tokyo on December 7, 2018 in Partial Fulfillment of the Requirements for the Degree of Doctor of Information Science and Technology in Computer Science ii Abstract Universal algebra uniformly captures various algebraic structures, by expressing them as equational theories or abstract clones. The ubiquity of algebraic structures in math- ematics and related fields has given rise to several variants of universal algebra, such as symmetric operads, non-symmetric operads, generalised operads, and monads. These variants of universal algebra are called notions of algebraic theory. Although notions of algebraic theory share the basic aim of providing a background theory to describe alge- braic structures, they use various techniques to achieve this goal and, to the best of our knowledge, no general framework for notions of algebraic theory which includes all of the examples above was known. Such a framework would lead to a better understand- ing of notions of algebraic theory by revealing their essential structure, and provide a uniform way to compare different notions of algebraic theory. In the first part of this thesis, we develop a unified framework for notions of algebraic theory which includes all of the above examples. Our key observation is that each notion of algebraic theory can be identified with a monoidal category, in such a way that theories correspond to monoid objects therein. We introduce a categorical structure called metamodel, which underlies the definition of models of theories. -
DEFINITIONS and THEOREMS in GENERAL TOPOLOGY 1. Basic
DEFINITIONS AND THEOREMS IN GENERAL TOPOLOGY 1. Basic definitions. A topology on a set X is defined by a family O of subsets of X, the open sets of the topology, satisfying the axioms: (i) ; and X are in O; (ii) the intersection of finitely many sets in O is in O; (iii) arbitrary unions of sets in O are in O. Alternatively, a topology may be defined by the neighborhoods U(p) of an arbitrary point p 2 X, where p 2 U(p) and, in addition: (i) If U1;U2 are neighborhoods of p, there exists U3 neighborhood of p, such that U3 ⊂ U1 \ U2; (ii) If U is a neighborhood of p and q 2 U, there exists a neighborhood V of q so that V ⊂ U. A topology is Hausdorff if any distinct points p 6= q admit disjoint neigh- borhoods. This is almost always assumed. A set C ⊂ X is closed if its complement is open. The closure A¯ of a set A ⊂ X is the intersection of all closed sets containing X. A subset A ⊂ X is dense in X if A¯ = X. A point x 2 X is a cluster point of a subset A ⊂ X if any neighborhood of x contains a point of A distinct from x. If A0 denotes the set of cluster points, then A¯ = A [ A0: A map f : X ! Y of topological spaces is continuous at p 2 X if for any open neighborhood V ⊂ Y of f(p), there exists an open neighborhood U ⊂ X of p so that f(U) ⊂ V . -
Categorical Models of Type Theory
Categorical models of type theory Michael Shulman February 28, 2012 1 / 43 Theories and models Example The theory of a group asserts an identity e, products x · y and inverses x−1 for any x; y, and equalities x · (y · z) = (x · y) · z and x · e = x = e · x and x · x−1 = e. I A model of this theory (in sets) is a particularparticular group, like Z or S3. I A model in spaces is a topological group. I A model in manifolds is a Lie group. I ... 3 / 43 Group objects in categories Definition A group object in a category with finite products is an object G with morphisms e : 1 ! G, m : G × G ! G, and i : G ! G, such that the following diagrams commute. m×1 (e;1) (1;e) G × G × G / G × G / G × G o G F G FF xx 1×m m FF xx FF m xx 1 F x 1 / F# x{ x G × G m G G ! / e / G 1 GO ∆ m G × G / G × G 1×i 4 / 43 Categorical semantics Categorical semantics is a general procedure to go from 1. the theory of a group to 2. the notion of group object in a category. A group object in a category is a model of the theory of a group. Then, anything we can prove formally in the theory of a group will be valid for group objects in any category. 5 / 43 Doctrines For each kind of type theory there is a corresponding kind of structured category in which we consider models. -
Arxiv:1909.05807V1 [Math.RA] 12 Sep 2019 Cs T
MODULES OVER TRUSSES VS MODULES OVER RINGS: DIRECT SUMS AND FREE MODULES TOMASZ BRZEZINSKI´ AND BERNARD RYBOLOWICZ Abstract. Categorical constructions on heaps and modules over trusses are con- sidered and contrasted with the corresponding constructions on groups and rings. These include explicit description of free heaps and free Abelian heaps, coprod- ucts or direct sums of Abelian heaps and modules over trusses, and description and analysis of free modules over trusses. It is shown that the direct sum of two non-empty Abelian heaps is always infinite and isomorphic to the heap associated to the direct sums of the group retracts of both heaps and Z. Direct sum is used to extend a given truss to a ring-type truss or a unital truss (or both). Free mod- ules are constructed as direct sums of a truss. It is shown that only free rank-one modules are free as modules over the associated truss. On the other hand, if a (finitely generated) module over a truss associated to a ring is free, then so is the corresponding quotient-by-absorbers module over this ring. 1. Introduction Trusses and skew trusses were defined in [3] in order to capture the nature of the distinctive distributive law that characterises braces and skew braces [12], [6], [9]. A (one-sided) truss is a set with a ternary operation which makes it into a heap or herd (see [10], [11], [1] or [13]) together with an associative binary operation that distributes (on one side or both) over the heap ternary operation. If the specific bi- nary operation admits it, a choice of a particular element could fix a group structure on a heap in a way that turns the truss into a ring or a brace-like system (which becomes a brace provided the binary operation admits inverses). -
An Outline of Algebraic Set Theory
An Outline of Algebraic Set Theory Steve Awodey Dedicated to Saunders Mac Lane, 1909–2005 Abstract This survey article is intended to introduce the reader to the field of Algebraic Set Theory, in which models of set theory of a new and fascinating kind are determined algebraically. The method is quite robust, admitting adjustment in several respects to model different theories including classical, intuitionistic, bounded, and predicative ones. Under this scheme some familiar set theoretic properties are related to algebraic ones, like freeness, while others result from logical constraints, like definability. The overall theory is complete in two important respects: conventional elementary set theory axiomatizes the class of algebraic models, and the axioms provided for the abstract algebraic framework itself are also complete with respect to a range of natural models consisting of “ideals” of sets, suitably defined. Some previous results involving realizability, forcing, and sheaf models are subsumed, and the prospects for further such unification seem bright. 1 Contents 1 Introduction 3 2 The category of classes 10 2.1 Smallmaps ............................ 12 2.2 Powerclasses............................ 14 2.3 UniversesandInfinity . 15 2.4 Classcategories .......................... 16 2.5 Thetoposofsets ......................... 17 3 Algebraic models of set theory 18 3.1 ThesettheoryBIST ....................... 18 3.2 Algebraic soundness of BIST . 20 3.3 Algebraic completeness of BIST . 21 4 Classes as ideals of sets 23 4.1 Smallmapsandideals . .. .. 24 4.2 Powerclasses and universes . 26 4.3 Conservativity........................... 29 5 Ideal models 29 5.1 Freealgebras ........................... 29 5.2 Collection ............................. 30 5.3 Idealcompleteness . .. .. 32 6 Variations 33 References 36 2 1 Introduction Algebraic set theory (AST) is a new approach to the construction of models of set theory, invented by Andr´eJoyal and Ieke Moerdijk and first presented in [16].