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ON Hg-HOMEOMORPHISMS in TOPOLOGICAL SPACES
Proyecciones Vol. 28, No 1, pp. 1—19, May 2009. Universidad Cat´olica del Norte Antofagasta - Chile ON g-HOMEOMORPHISMS IN TOPOLOGICAL SPACES e M. CALDAS UNIVERSIDADE FEDERAL FLUMINENSE, BRASIL S. JAFARI COLLEGE OF VESTSJAELLAND SOUTH, DENMARK N. RAJESH PONNAIYAH RAMAJAYAM COLLEGE, INDIA and M. L. THIVAGAR ARUL ANANDHAR COLLEGE Received : August 2006. Accepted : November 2008 Abstract In this paper, we first introduce a new class of closed map called g-closed map. Moreover, we introduce a new class of homeomor- phism called g-homeomorphism, which are weaker than homeomor- phism.e We prove that gc-homeomorphism and g-homeomorphism are independent.e We also introduce g*-homeomorphisms and prove that the set of all g*-homeomorphisms forms a groupe under the operation of composition of maps. e e 2000 Math. Subject Classification : 54A05, 54C08. Keywords and phrases : g-closed set, g-open set, g-continuous function, g-irresolute map. e e e e 2 M. Caldas, S. Jafari, N. Rajesh and M. L. Thivagar 1. Introduction The notion homeomorphism plays a very important role in topology. By definition, a homeomorphism between two topological spaces X and Y is a 1 bijective map f : X Y when both f and f − are continuous. It is well known that as J¨anich→ [[5], p.13] says correctly: homeomorphisms play the same role in topology that linear isomorphisms play in linear algebra, or that biholomorphic maps play in function theory, or group isomorphisms in group theory, or isometries in Riemannian geometry. In the course of generalizations of the notion of homeomorphism, Maki et al. -
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. -
Open 3-Manifolds Which Are Simply Connected at Infinity
OPEN 3-MANIFOLDS WHICH ARE SIMPLY CONNECTED AT INFINITY C H. EDWARDS, JR.1 A triangulated open manifold M will be called l-connected at infin- ity il each compact subset C of M is contained in a compact poly- hedron P in M such that M—P is connected and simply connected. Stallings has shown that, if M is a contractible open combinatorial manifold which is l-connected at infinity and is of dimension ra^5, then M is piecewise-linearly homeomorphic to Euclidean re-space E" [5]. Theorem 1. Let M be a contractible open 3-manifold, each of whose compact subsets can be imbedded in £3. If M is l-connected at infinity, then M is homeomorphic to £3. Notice that, in order to prove the 3-dimensional Poincaré conjec- ture, it would suffice to prove Theorem 1 without the hypothesis that each compact subset of M can be imbedded in £3. For, if M is a simply connected closed 3-manifold and p is a point of M, then M —p is a contractible open 3-manifold which is clearly l-connected at infinity. Conversely, if the 3-dimensional Poincaré conjecture were known, then the hypothesis that each compact subset of M can be imbedded in £3 would be unnecessary. All spaces and mappings in this paper are considered in the poly- hedral or piecewise-linear sense, unless otherwise stated. As usual, by an open re-manifold is meant a noncompact connected space triangu- lated by a countable simplicial complex without boundary, such that the link of each vertex is piecewise-linearly homeomorphic to the usual (re—1)-sphere. -
Cohomological Descent on the Overconvergent Site
COHOMOLOGICAL DESCENT ON THE OVERCONVERGENT SITE DAVID ZUREICK-BROWN Abstract. We prove that cohomological descent holds for finitely presented crystals on the overconvergent site with respect to proper or fppf hypercovers. 1. Introduction Cohomological descent is a robust computational and theoretical tool, central to p-adic cohomology and its applications. On one hand, it facilitates explicit calculations (analo- gous to the computation of coherent cohomology in scheme theory via Cechˇ cohomology); on another, it allows one to deduce results about singular schemes (e.g., finiteness of the cohomology of overconvergent isocrystals on singular schemes [Ked06]) from results about smooth schemes, and, in a pinch, sometimes allows one to bootstrap global definitions from local ones (for example, for a scheme X which fails to embed into a formal scheme smooth near X, one actually defines rigid cohomology via cohomological descent; see [lS07, comment after Proposition 8.2.17]). The main result of the series of papers [CT03,Tsu03,Tsu04] is that cohomological descent for the rigid cohomology of overconvergent isocrystals holds with respect to both flat and proper hypercovers. The barrage of choices in the definition of rigid cohomology is burden- some and makes their proofs of cohomological descent very difficult, totaling to over 200 pages. Even after the main cohomological descent theorems [CT03, Theorems 7.3.1 and 7.4.1] are proved one still has to work a bit to get a spectral sequence [CT03, Theorem 11.7.1]. Actually, even to state what one means by cohomological descent (without a site) is subtle. The situation is now more favorable. -
Homological Algebra
Homological Algebra Donu Arapura April 1, 2020 Contents 1 Some module theory3 1.1 Modules................................3 1.6 Projective modules..........................5 1.12 Projective modules versus free modules..............7 1.15 Injective modules...........................8 1.21 Tensor products............................9 2 Homology 13 2.1 Simplicial complexes......................... 13 2.8 Complexes............................... 15 2.15 Homotopy............................... 18 2.23 Mapping cones............................ 19 3 Ext groups 21 3.1 Extensions............................... 21 3.11 Projective resolutions........................ 24 3.16 Higher Ext groups.......................... 26 3.22 Characterization of projectives and injectives........... 28 4 Cohomology of groups 32 4.1 Group cohomology.......................... 32 4.6 Bar resolution............................. 33 4.11 Low degree cohomology....................... 34 4.16 Applications to finite groups..................... 36 4.20 Topological interpretation...................... 38 5 Derived Functors and Tor 39 5.1 Abelian categories.......................... 39 5.13 Derived functors........................... 41 5.23 Tor functors.............................. 44 5.28 Homology of a group......................... 45 1 6 Further techniques 47 6.1 Double complexes........................... 47 6.7 Koszul complexes........................... 49 7 Applications to commutative algebra 52 7.1 Global dimensions.......................... 52 7.9 Global dimension of -
Sheaves and Homotopy Theory
SHEAVES AND HOMOTOPY THEORY DANIEL DUGGER The purpose of this note is to describe the homotopy-theoretic version of sheaf theory developed in the work of Thomason [14] and Jardine [7, 8, 9]; a few enhancements are provided here and there, but the bulk of the material should be credited to them. Their work is the foundation from which Morel and Voevodsky build their homotopy theory for schemes [12], and it is our hope that this exposition will be useful to those striving to understand that material. Our motivating examples will center on these applications to algebraic geometry. Some history: The machinery in question was invented by Thomason as the main tool in his proof of the Lichtenbaum-Quillen conjecture for Bott-periodic algebraic K-theory. He termed his constructions `hypercohomology spectra', and a detailed examination of their basic properties can be found in the first section of [14]. Jardine later showed how these ideas can be elegantly rephrased in terms of model categories (cf. [8], [9]). In this setting the hypercohomology construction is just a certain fibrant replacement functor. His papers convincingly demonstrate how many questions concerning algebraic K-theory or ´etale homotopy theory can be most naturally understood using the model category language. In this paper we set ourselves the specific task of developing some kind of homotopy theory for schemes. The hope is to demonstrate how Thomason's and Jardine's machinery can be built, step-by-step, so that it is precisely what is needed to solve the problems we encounter. The papers mentioned above all assume a familiarity with Grothendieck topologies and sheaf theory, and proceed to develop the homotopy-theoretic situation as a generalization of the classical case. -
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)). -
Mathematics 1
Mathematics 1 Mathematics Department Information • Department Chair: Friedrich Littmann, Ph.D. • Graduate Coordinator: Indranil Sengupta, Ph.D. • Department Location: 412 Minard Hall • Department Phone: (701) 231-8171 • Department Web Site: www.ndsu.edu/math (http://www.ndsu.edu/math/) • Application Deadline: March 1 to be considered for assistantships for fall. Openings may be very limited for spring. • Credential Offered: Ph.D., M.S. • English Proficiency Requirements: TOEFL iBT 71; IELTS 6 Program Description The Department of Mathematics offers graduate study leading to the degrees of Master of Science (M.S.) and Doctor of Philosophy (Ph.D.). Advanced work may be specialized among the following areas: • algebra, including algebraic number theory, commutative algebra, and homological algebra • analysis, including analytic number theory, approximation theory, ergodic theory, harmonic analysis, and operator algebras • applied mathematics, mathematical finance, mathematical biology, differential equations, dynamical systems, • combinatorics and graph theory • geometry/topology, including differential geometry, geometric group theory, and symplectic topology Beginning with their first year in residence, students are strongly urged to attend research seminars and discuss research opportunities with faculty members. By the end of their second semester, students select an advisory committee and develop a plan of study specifying how all degree requirements are to be met. One philosophical tenet of the Department of Mathematics graduate program is that each mathematics graduate student will be well grounded in at least two foundational areas of mathematics. To this end, each student's background will be assessed, and the student will be directed to the appropriate level of study. The Department of Mathematics graduate program is open to all qualified graduates of universities and colleges of recognized standing. -
Professor Smith Math 295 Lecture Notes
Professor Smith Math 295 Lecture Notes by John Holler Fall 2010 1 October 29: Compactness, Open Covers, and Subcovers. 1.1 Reexamining Wednesday's proof There seemed to be a bit of confusion about our proof of the generalized Intermediate Value Theorem which is: Theorem 1: If f : X ! Y is continuous, and S ⊆ X is a connected subset (con- nected under the subspace topology), then f(S) is connected. We'll approach this problem by first proving a special case of it: Theorem 1: If f : X ! Y is continuous and surjective and X is connected, then Y is connected. Proof of 2: Suppose not. Then Y is not connected. This by definition means 9 non-empty open sets A; B ⊂ Y such that A S B = Y and A T B = ;. Since f is continuous, both f −1(A) and f −1(B) are open. Since f is surjective, both f −1(A) and f −1(B) are non-empty because A and B are non-empty. But the surjectivity of f also means f −1(A) S f −1(B) = X, since A S B = Y . Additionally we have f −1(A) T f −1(B) = ;. For if not, then we would have that 9x 2 f −1(A) T f −1(B) which implies f(x) 2 A T B = ; which is a contradiction. But then these two non-empty open sets f −1(A) and f −1(B) disconnect X. QED Now we want to show that Theorem 2 implies Theorem 1. This will require the help from a few lemmas, whose proofs are on Homework Set 7: Lemma 1: If f : X ! Y is continuous, and S ⊆ X is any subset of X then the restriction fjS : S ! Y; x 7! f(x) is also continuous. -
07. Homeomorphisms and Embeddings
07. Homeomorphisms and embeddings Homeomorphisms are the isomorphisms in the category of topological spaces and continuous functions. Definition. 1) A function f :(X; τ) ! (Y; σ) is called a homeomorphism between (X; τ) and (Y; σ) if f is bijective, continuous and the inverse function f −1 :(Y; σ) ! (X; τ) is continuous. 2) (X; τ) is called homeomorphic to (Y; σ) if there is a homeomorphism f : X ! Y . Remarks. 1) Being homeomorphic is apparently an equivalence relation on the class of all topological spaces. 2) It is a fundamental task in General Topology to decide whether two spaces are homeomorphic or not. 3) Homeomorphic spaces (X; τ) and (Y; σ) cannot be distinguished with respect to their topological structure because a homeomorphism f : X ! Y not only is a bijection between the elements of X and Y but also yields a bijection between the topologies via O 7! f(O). Hence a property whose definition relies only on set theoretic notions and the concept of an open set holds if and only if it holds in each homeomorphic space. Such properties are called topological properties. For example, ”first countable" is a topological property, whereas the notion of a bounded subset in R is not a topological property. R ! − t Example. The function f : ( 1; 1) with f(t) = 1+jtj is a −1 x homeomorphism, the inverse function is f (x) = 1−|xj . 1 − ! b−a a+b The function g :( 1; 1) (a; b) with g(x) = 2 x + 2 is a homeo- morphism. Therefore all open intervals in R are homeomorphic to each other and homeomorphic to R . -
Mathematics (MATH)
Course Descriptions MATH 1080 QL MATH 1210 QL Mathematics (MATH) Precalculus Calculus I 5 5 MATH 100R * Prerequisite(s): Within the past two years, * Prerequisite(s): One of the following within Math Leap one of the following: MAT 1000 or MAT 1010 the past two years: (MATH 1050 or MATH 1 with a grade of B or better or an appropriate 1055) and MATH 1060, each with a grade of Is part of UVU’s math placement process; for math placement score. C or higher; OR MATH 1080 with a grade of C students who desire to review math topics Is an accelerated version of MATH 1050 or higher; OR appropriate placement by math in order to improve placement level before and MATH 1060. Includes functions and placement test. beginning a math course. Addresses unique their graphs including polynomial, rational, Covers limits, continuity, differentiation, strengths and weaknesses of students, by exponential, logarithmic, trigonometric, and applications of differentiation, integration, providing group problem solving activities along inverse trigonometric functions. Covers and applications of integration, including with an individual assessment and study inequalities, systems of linear and derivatives and integrals of polynomial plan for mastering target material. Requires nonlinear equations, matrices, determinants, functions, rational functions, exponential mandatory class attendance and a minimum arithmetic and geometric sequences, the functions, logarithmic functions, trigonometric number of hours per week logged into a Binomial Theorem, the unit circle, right functions, inverse trigonometric functions, and preparation module, with progress monitored triangle trigonometry, trigonometric equations, hyperbolic functions. Is a prerequisite for by a mentor. May be repeated for a maximum trigonometric identities, the Law of Sines, the calculus-based sciences. -
Lecture 15. De Rham Cohomology
Lecture 15. de Rham cohomology In this lecture we will show how differential forms can be used to define topo- logical invariants of manifolds. This is closely related to other constructions in algebraic topology such as simplicial homology and cohomology, singular homology and cohomology, and Cechˇ cohomology. 15.1 Cocycles and coboundaries Let us first note some applications of Stokes’ theorem: Let ω be a k-form on a differentiable manifold M.For any oriented k-dimensional compact sub- manifold Σ of M, this gives us a real number by integration: " ω : Σ → ω. Σ (Here we really mean the integral over Σ of the form obtained by pulling back ω under the inclusion map). Now suppose we have two such submanifolds, Σ0 and Σ1, which are (smoothly) homotopic. That is, we have a smooth map F : Σ × [0, 1] → M with F |Σ×{i} an immersion describing Σi for i =0, 1. Then d(F∗ω)isa (k + 1)-form on the (k + 1)-dimensional oriented manifold with boundary Σ × [0, 1], and Stokes’ theorem gives " " " d(F∗ω)= ω − ω. Σ×[0,1] Σ1 Σ1 In particular, if dω =0,then d(F∗ω)=F∗(dω)=0, and we deduce that ω = ω. Σ1 Σ0 This says that k-forms with exterior derivative zero give a well-defined functional on homotopy classes of compact oriented k-dimensional submani- folds of M. We know some examples of k-forms with exterior derivative zero, namely those of the form ω = dη for some (k − 1)-form η. But Stokes’ theorem then gives that Σ ω = Σ dη =0,sointhese cases the functional we defined on homotopy classes of submanifolds is trivial.