DOCSLIB.ORG

Introduction to Topological Groups

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Introduction to Topological Groups Topologia 2, 2017/18 Topological Groups Versione 26.02.2018 Introduction to Topological Groups Dikran Dikranjan To the memory of Ivan Prodanov (1935 { 1985) Abstract These notes provide a brief introduction to topological groups with a special emphasis on Pontryagin-van Kam- pen's duality theorem for locally compact abelian groups. We give a completely self-contained elementary proof of the theorem following the line from [57, 67]. According to the classical tradition, the structure theory of the locally compact abelian groups is built parallelly. 1 Introduction Let L denote the category of locally compact abelian groups and continuous homomorphisms and let T = R=Z be the unit circle group. For G 2 L denote by Gb the group of continuous homomorphisms (characters) G ! T equipped with the compact-open topology. Then the assignment G 7! Gb is a contravariant endofunctor b: L!L. The celebrated Pontryagin-van Kampen duality theorem ([122]) says that this functor is, up to natural equivalence, an involution i.e., Gb =∼ G and this isomorphism is \well behaved" (see Theorem 12.5.4 for more detail). Moreover, this functor sends compact groups to discrete ones and viceversa, i.e., it defines a duality between the subcategory C of compact abelian groups and the subcategory D of discrete abelian groups. This allows for a very efficient and fruitful tool for the study of compact abelian groups, reducing many problems related to topological properties of these group to the related problems concerning algebraic properties in the category of discrete groups. The reader is advised to give a look at the Mackey's beautiful survey [114] for the connection of charactres and Pontryagin-van Kampen duality to number theory, physics and elsewhere. This duality inspired a huge amount of related research also in category theory, a brief comment on a specific categorical aspect (uniqueness and representability) can be found in x8.1 of the Appendix. The aim of these notes is to provide a self-contained proof of this remarkable duality theorem, providing all necessary steps, including basic background on topological groups and the structure theory of locally compact abelian groups. Peter-Weyl's theorem asserting that the continuous characters of the compact abelian groups separate the points of the groups (see Theorem 11.2.1) is certainly the most important tool in proving the duality theorem. The Peter-Weyl's theorem is valid for arbitrary compact groups, but then continuous characters must be replaced by finite-dimensional unitary representations and the usual proof of the theorem in this general case involves Haar integration. In the case of abelian groups the irreducible ones turn out the be one-dimensional, i.e., continuous characters. We prefer here a different approach. Namely, Peter-Weyl's theorem in the abelian case can be obtained as an immediate corollary of a theorem of Følner (Theorem 10.3.5) whose elementary proof uses nothing beyond elementary properties of the finite abelian groups, a local version of the Stone-Weierstraß approximation theorem proved in x2 and the Stone-Cechˇ compactification of discrete spaces. As another application of Følner’s theorem we describe the precompact groups (i.e., the subgroups of the compact groups) as having a topology generated by continuous characters. As a third application of Følner’s theorem one can obtain the existence of the Haar integral on locally compact abelian groups for free (see Theorem 11.4.21, the proof follows [57, x2.4, Theorem 2.4.5]). The notes are organized as follows. In Section 2 we recall basic results and notions on abelian groups and general topology, which will be used in the rest of the paper. Section 3 contains background on topological groups, starting from scratch. Various ways of introducing a group topology are considered (x3.2), of which the prominent one is by means of characters (x3.2.3). In x4.3 we recall the construction of Protasov and Zelenyuk [131] of topologies arising from a given sequence that is required to be convergent to 0. In x4.1 we discuss separation axioms and metrizability of topological groups. Connectedness and related properties in topological groups are discussed in x4.2. In x5 the Markov's problems on the existence of non-discrete Hausdorff group topologies is discussed. In x5.1 we introduce two topologies, the Markov topology and the Zariski topology, that allow for an easier understanding of i Markov's problems. In x5.2 we describe the Markov topology of the infinite permutation groups, while x5.3 contains the first two examples of non-topologizable groups, given by Shelah and Ol0shanskii, respectively. The problems arising in extension of group topologies are the topic of x5.4. Several cardinal invariants (weight, character and density character) are introduced in x6.1, whereas x6.2 discuses completeness and completions. Further general information on topological groups can be found in the monographs or surveys [4, 36, 37, 38, 57, 106, 119, 122]. Section 7 is dedicated to specific properties of the (locally) compact groups used essentially in these notes. The most important property of locally compact group we recall in x7.3 is the open mapping theorem. x7.4 is dedicated to the minimal and the totally minimal groups, which need not be locally compact , yet satisfy the open mapping theorem. In x8 we recall (with complete proofs) the structure of the closed subgroups of Rn as well as the description of the closure of an arbitrary subgroup of Rn. These groups play an important role in the whole theory of locally compact abelian groups. Section 9 starts with x9.1 dedicated to big (large) and small subsets of abstract groups. In x9.2.1 we give an internal description of the precompact groups using the notion of a big set of a group and we show that these are precisely the subgroups of the compact groups. Moreover, we define a precompact group G+ that \best approximates" G. Its completion bG, the Bohr compactification of G, is the compact group that \best approximates" G. Here we introduce almost periodic functions and briefly comment their connection to the Bohr compactification of G. In x9.2.2 we establish the precompactness of the topologies generated by characters. In x9.3 we recall (without proofs) some relevant notions in the non-abelian case, as Haar integral, unitary representation, etc., that play a prominent role in the general theory of topological groups, but are not used in this exposition. The Haar integral in locally compact abelian groups is built in x11.4.3. In x10 prepares all ingredients for the proof of Følner’s theorem (see Theorem 10.3.5). This proof, follows the line of [57]. An important feature of the proof is the crucial idea, due to Prodanov, to eliminate all discontinuous characters in the uniform approximation of continuous functions via linear combinations of characters obtained by means of Stone- Weierstraß approximation theorem. This step is ensured by Prodanov's lemma 10.3.1, which has many other relevant applications towards independence of characters and the construction of the Haar intergral for LCA groups. The last subsection contains the final stage of the proof of Følner’s theorem. In Section 11 gives various applications of Følner’s theorem. The first one is a description of the precompact group topologies of the abelian groups (x10.1). The main application of Følner’s theorem is an immediate proof of Peter- Weyl's theorem (in x11.2). To the structure of the compactly generated locally compact abelian groups is dedicated x11.3. Applications of these structure theorems are given in x11.4.3. In x11.3 we provide useful information on the dual of a locally compact abelian group, to be used in Section 12. x11.4 is dedicated to the almost periodic functions of the abelian group. As another application of Følner's theorem we give a proof of Bohr - von Neumann's theorem describing the almost periodic functions as uniform limits of linear combinations of characters. Among other things, we obtain as a by-product of Prodanov's approach an easy construction of the Haar integral for almost periodic functions on abelian groups, in particular for all continuous functions on a compact abelian group (x11.4.2). In x11.4.3 we build a Haar integral on arbitrary locally compact abelian groups, using the construction from x11.4.2 in the compact case. In x11.5 we consider a precompact version of the construction form x4.3 of topologies making a fixed sequence converge to 0. Section 12 is dedicated to Pontryagin-van Kampen duality. In xx12.1-12.3 we construct all tools for proving the duality theorem 12.5.4. More specifically, xx12.1 and 12.2 contain various properties of the dual groups that allow for an easier computation of the dual in many cases. Using further the properties of the dual, we see in x12.3 that many specific groups G satisfy the duality theorem, i.e., G =∼ Gb. In x12.4 we stress the fact that the isomorphism G =∼ Gb is natural by studying in detail the natural transformation !G : G ! Gb connecting the group with its bidual. It is shown in several steps that !G is an isomorphism, considering larger and larger classes of locally compact abelian groups G where the duality theorem holds (elementary locally compact abelian groups, compact abelian groups, discrete abelian groups, compactly generated locally compact abelian groups). The last step uses the fact that the duality functor is exact, this permits us to use all previous steps in the general case. As an immediate application of the duality theorem we obtain the main structure theorem for the locally compact abelian groups, a complete description of the monothetic compact groups, the torsion compact abelian groups, the connected compact abelian groups with dense torsion subgroup, etc.
Load more

Recommended publications
  • On S-Topological Groups
    Mathematica Moravica Vol. 18-2 (2014), 35–44 On s-Topological Groups Muhammad Siddique Bosan, Moiz ud Din Khan and Ljubiša D.R. Kočinac∗) Abstract. In this paper we study the class of s-topological groups and a wider class of S-topological groups which are defined by us- ing semi-open sets and semi-continuity introduced by N. Levine. It is shown that these groups form a generalization of topological groups, and that they are different from several distinct notions of semitopolog- ical groups which appear in the literature. Counterexamples are given to strengthen these concepts. Some basic results and applications of s- and S-topological groups are presented. Similarities with and differ- ences from topological groups are investigated. 1. Introduction If a set is endowed with algebraic and topological structures, then it is natural to consider and investigate interplay between these two structures. The most natural way for such a study is to require algebraic operations to be continuous. It is the case in investigation of topological groups: the multi- plication mapping and the inverse mapping are continuous. Similar situation is with topological rings, topological vector spaces and so on. However, it is also natural to see what will happen if some of algebraic operations sat- isfy certain weaker forms of continuity. Such a study in connection with topological groups started in the 1930s and 1950s and led to investigation of semitopological groups (the multiplication mapping is separately con- tinuous), paratopological groups (the multiplication is jointly continuous), quasi-topological groups (which are semitopological groups with continu- ous inverse mapping).
    [Show full text]
  • Dynamics for Discrete Subgroups of Sl 2(C)
    DYNAMICS FOR DISCRETE SUBGROUPS OF SL2(C) HEE OH Dedicated to Gregory Margulis with affection and admiration Abstract. Margulis wrote in the preface of his book Discrete subgroups of semisimple Lie groups [30]: \A number of important topics have been omitted. The most significant of these is the theory of Kleinian groups and Thurston's theory of 3-dimensional manifolds: these two theories can be united under the common title Theory of discrete subgroups of SL2(C)". In this article, we will discuss a few recent advances regarding this missing topic from his book, which were influenced by his earlier works. Contents 1. Introduction 1 2. Kleinian groups 2 3. Mixing and classification of N-orbit closures 10 4. Almost all results on orbit closures 13 5. Unipotent blowup and renormalizations 18 6. Interior frames and boundary frames 25 7. Rigid acylindrical groups and circular slices of Λ 27 8. Geometrically finite acylindrical hyperbolic 3-manifolds 32 9. Unipotent flows in higher dimensional hyperbolic manifolds 35 References 44 1. Introduction A discrete subgroup of PSL2(C) is called a Kleinian group. In this article, we discuss dynamics of unipotent flows on the homogeneous space Γn PSL2(C) for a Kleinian group Γ which is not necessarily a lattice of PSL2(C). Unlike the lattice case, the geometry and topology of the associated hyperbolic 3-manifold M = ΓnH3 influence both topological and measure theoretic rigidity properties of unipotent flows. Around 1984-6, Margulis settled the Oppenheim conjecture by proving that every bounded SO(2; 1)-orbit in the space SL3(Z)n SL3(R) is compact ([28], [27]).
    [Show full text]
  • Functorial Constructions in Paratopological Groups Reflecting Separation Axioms
    Functorial constructions in paratopological groups reflecting separation axioms Mikhail Tkachenko Universidad Aut´onomaMetropolitana, Mexico City [email protected] Brazilian Conference on General Topology and Set Theory S~aoSebasti~ao,Brazil, 2013 In honor of Ofelia T. Alas Contents: 1. Three known functorial constructions 2. Each axiom of separation has its functorial reflection 3. `Internal' description of the groups Tk (G) 4. Properties of the functors Tk 's 5. Products and functors 6. Some applications A paratopological group is a group G with topology such that multiplication in G is jointly continuous. `topological' =) `paratopological' =) `semitopological' Let (G; τ) be a paratopological group and τ −1 = fU−1 : U 2 τg be the conjugate topology of G. Then G 0 = (G; τ −1) is also a paratopological group and the inversion in G is a homeomorphism of (G; τ) onto (G; τ −1). Let τ ∗ = τ _ τ −1 be the least upper bound of τ and τ −1. Then G ∗ = (G; τ ∗) is a topological group associated to G. ∗ For the Sorgenfrey line S, the topological group S is discrete. Paratopological and semitopological groups A semitopological group is an abstract group G with topology τ such that the left and right translations in G are continuous or, equivalently, multiplication in G is separately continuous. `topological' =) `paratopological' =) `semitopological' Let (G; τ) be a paratopological group and τ −1 = fU−1 : U 2 τg be the conjugate topology of G. Then G 0 = (G; τ −1) is also a paratopological group and the inversion in G is a homeomorphism of (G; τ) onto (G; τ −1).
    [Show full text]
  • Syllabus for M.Phil
    Structure of Syllabus for M.Phil. in Mathematics Semester-I Full Full Full Full Course Course Code Course Name Marks Marks Marks Marks Credit Type (External) (Practical) (Internal) (Total) Research Methodology: MPHILMA-101 Theory Research Foundation 20 5 25 2 Research Methodology: Computer Application in MPHILMA-102 Practical 20 5 25 2 Research Elective: Any two papers to be chosen from Table-I, 40 10 50 4 based on research interest of MPHILMA-103 Theory + + + + the students and availability of suitable teachers/ 40 10 50 4 Supervisors. Total 120 30 150 12 Semester-II Full Full Full Full Course Course Code Course Name Marks Marks Marks Marks Credit Type (External) (Practical) (Internal) (Total) Elective: Any three papers to 40 10 50 4 be chosen from Table-II, + + + + MPHILMA-201 Theory based on research interest of 40 10 50 4 the students and availability of + + + + suitable teachers/Supervisors. 40 10 50 4 Total 120 30 150 12 Semester-III Full Full Full Full Course Course Code Course Name Marks Marks Marks Marks Credit Type (External) (Practical) (Internal) (Total) Preliminary MPHILMA-301 Theory 75 75 6 Dissertation MPHILMA-302 Theory Viva-voce 25 25 2 Total 100 100 8 Semester-IV Full Full Full Full Course Course Code Course Name Marks Marks Marks Marks Credit Type (External) (Practical) (Internal) (Total) MPHILMA-401 Theory Final Dissertation 75 75 6 MPHILMA-402 Theory Viva-voce 25 25 2 Total 100 100 8 Table: I Elective Papers for MPHILMA-103 (M1) to MPHILMA-103 (M18) Elective Paper Title of the Paper Sub-Code M1 Theory of Convergence
    [Show full text]
  • Counterexamples in Topology
    Counterexamples in Topology Lynn Arthur Steen Professor of Mathematics, Saint Olaf College and J. Arthur Seebach, Jr. Professor of Mathematics, Saint Olaf College DOVER PUBLICATIONS, INC. New York Contents Part I BASIC DEFINITIONS 1. General Introduction 3 Limit Points 5 Closures and Interiors 6 Countability Properties 7 Functions 7 Filters 9 2. Separation Axioms 11 Regular and Normal Spaces 12 Completely Hausdorff Spaces 13 Completely Regular Spaces 13 Functions, Products, and Subspaces 14 Additional Separation Properties 16 3. Compactness 18 Global Compactness Properties 18 Localized Compactness Properties 20 Countability Axioms and Separability 21 Paracompactness 22 Compactness Properties and Ts Axioms 24 Invariance Properties 26 4. Connectedness 28 Functions and Products 31 Disconnectedness 31 Biconnectedness and Continua 33 VII viii Contents 5. Metric Spaces 34 Complete Metric Spaces 36 Metrizability 37 Uniformities 37 Metric Uniformities 38 Part II COUNTEREXAMPLES 1. Finite Discrete Topology 41 2. Countable Discrete Topology 41 3. Uncountable Discrete Topology 41 4. Indiscrete Topology 42 5. Partition Topology 43 6. Odd-Even Topology 43 7. Deleted Integer Topology 43 8. Finite Particular Point Topology 44 9. Countable Particular Point Topology 44 10. Uncountable Particular Point Topology 44 11. Sierpinski Space 44 12. Closed Extension Topology 44 13. Finite Excluded Point Topology 47 14. Countable Excluded Point Topology 47 15. Uncountable Excluded Point Topology 47 16. Open Extension Topology 47 17. Either-Or Topology 48 18. Finite Complement Topology on a Countable Space 49 19. Finite Complement Topology on an Uncountable Space 49 20. Countable Complement Topology 50 21. Double Pointed Countable Complement Topology 50 22. Compact Complement Topology 51 23.
    [Show full text]
  • Some Groups of Finite Homological Type
    Some groups of finite homological type Ian J. Leary∗ M¨ugeSaadeto˘glu† June 10, 2005 Abstract For each n ≥ 0 we construct a torsion-free group that satisfies K. S. Brown’s FHT condition and is FPn, but is not FPn+1. 1 Introduction While working on comparing different notions of Euler characteristic, K. S. Brown introduced a new homological finiteness condition for discrete groups [5, IX.6]. The group G is said to be of finite homological type or FHT if G has finite virtual cohomological dimension, and for every G-module M whose underlying abelian group is finitely generated, the homology groups Hi(G; M) are all finitely generated. If G is FHT , then one may define a ‘na¨ıve Euler characteristic’ for every finite-index subgroup H of G, as the alternating sum of the dimensions of the homology groups of H with rational coefficients. One question that arises is the connection between FHT and the usual homological finiteness conditions FP and FPn, which were introduced by J.-P. Serre [8]. (We shall define these conditions below.) It is easy to see that any group G of type FP is FHT , and one might conjecture that every torsion-free group that is FHT is also of type FP . The aim of this paper is to show that this is not the case. For each n ≥ 0, we exhibit a torsion-free group Gn that is FHT and of type FPn, but that is not of type FPn+1. Our construction is based on R. Bieri’s construction of a group that is FPn but not FPn+1 [2, Prop 2.14].
    [Show full text]
  • A Theorem on Discrete Groups and Some Consequences of Kazdan's
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF FUNCTIONAL ANALYSIS 6, 203-207 (1970) A Theorem on Discrete Groups and Some Consequences of Kazdan’s Thesis ROBERT R. KALLMAN* Department of Mathematics, Yale University, New Haven, Connecticut 06520 Communicated by Irving Segal Received June 21, 1969 Let G be a noncompact simple Lie group, and let r be a discrete subgroup of G such that G/P has finite volume. The main result of this paper is that the left ring of P is a Type II1 von Neumann algebra. This result in turn is applied to solve, in some generality, a problem in dynamical systems posed by I. M. Gelfand. INTRODUCTION If G is a locally compact unimodular group, let Y(G) be the von Neumann algebra generated by left translations L(a) (a in G) on L2(G). Let I’ be a discrete subgroup of G, and let p be a measure on G/r which is invariant under the natural action of G. The measure class of p is uniquely determined. If f is in L2(G/F), let (V4f)W = flu-’ - 4. The following problem is of interest in dynamical systems (cf. Gelfand [3]). Let &?(G, r) be the von Neumann algebra on L2(G) @ L2(G/.F) generated by L(a) @ V(u) (u in G) and I @ M(4) (here 4 is in L”(G/r, p) and (&Z(+)f)(x) = ~$(x)f(x), for all f in L2(G/r)).
    [Show full text]
  • «Algebraic and Geometric Methods of Analysis»
    International scientific conference «Algebraic and geometric methods of analysis» Book of abstracts May 31 - June 5, 2017 Odessa Ukraine http://imath.kiev.ua/~topology/conf/agma2017/ LIST OF TOPICS • Algebraic methods in geometry • Differential geometry in the large • Geometry and topology of differentiable manifolds • General and algebraic topology • Dynamical systems and their applications • Geometric problems in mathematical analysis • Geometric and topological methods in natural sciences • History and methodology of teaching in mathematics ORGANIZERS • The Ministry of Education and Science of Ukraine • Odesa National Academy of Food Technologies • The Institute of Mathematics of the National Academy of Sciences of Ukraine • Taras Shevchenko National University of Kyiv • The International Geometry Center PROGRAM COMMITTEE Chairman: Prishlyak A. Maksymenko S. Rahula M. (Kyiv, Ukraine) (Kyiv, Ukraine) (Tartu, Estonia) Balan V. Matsumoto K. Sabitov I. (Bucharest, Romania) (Yamagata, Japan) (Moscow, Russia) Banakh T. Mashkov O. Savchenko A. (Lviv, Ukraine) (Kyiv, Ukraine) (Kherson, Ukraine) Fedchenko Yu. Mykytyuk I. Sergeeva А. (Odesa, Ukraine) (Lviv, Ukraine) (Odesa, Ukraine) Fomenko A. Milka A. Strikha M. (Moscow, Russia) (Kharkiv, Ukraine) (Kyiv, Ukraine) Fomenko V. Mikesh J. Shvets V. (Taganrog, Russia) (Olomouc, Czech Republic) (Odesa, Ukraine) Glushkov A. Mormul P. Shelekhov A. (Odesa, Ukraine) (Warsaw, Poland) (Tver, Russia) Haddad М. Moskaliuk S. Shurygin V. (Wadi al-Nasara, Syria) (Wien, Austri) (Kazan, Russia) Herega A. Panzhenskiy V. Vlasenko I. (Odesa, Ukraine) (Penza, Russia) (Kyiv, Ukraine) Khruslov E. Pastur L. Zadorozhnyj V. (Kharkiv, Ukraine) (Kharkiv, Ukraine) (Odesa, Ukraine) Kirichenko V. Plachta L. Zarichnyi M. (Moscow, Russia) (Krakov, Poland) (Lviv, Ukraine) Kirillov V. Pokas S. Zelinskiy Y. (Odesa, Ukraine) (Odesa, Ukraine) (Kyiv, Ukraine) Konovenko N.
    [Show full text]
  • Discrete Groups and Simple C*-Algebras 1 Introduction
    Discrete groups and simple C*-algebras by Erik Bedos* Department of Mathematics University of Oslo P.O.Box 1053, Blindern 0316 Oslo 3, Norway 1 Introduction Let G denote a discrete group and let us say that G is C* -simple if the reduced group C* -algebra associated with G is simple. We notice im­ mediately that there is no interest in considering here the full group C*­ algebra associated with G, because it is simple if and only if G is trivial. Since Powers in 1975 ([26]) proved that all non-abelian free groups are C*­ simple, the class of C* -simple groups has been considerably enlarged (see [1,2,6,7,12,13,14,16,24] as a sample!), and two important subclasses are the so-called weak Powers groups ([6,13]; see section 4 for definition and ex­ amples) and the groups of Akemann-Lee type ([1,2]), which are groups possessing a normal non-abelian free subgroup with trivial centralizer. The problem of giving an intrisic characterization of C* -simple groups is still open. It is known that a C* -simple group has no normal amenable subgroup other than the trivial one ([24; proposition 1.6]) and is ICC (since the center of the associated reduced group C* -algebra must be the scalars). One may of course wonder if the converse is true. On the other hand, most C* -simple groups are known to have a unique trace, i.e. the canonical trace on the reduced group C* -algebra· is unique, which naturally raises the problem whether this is always true or not ([13; §2, question (2)]).
    [Show full text]
  • Discreteness Properties of Translation Numbers in Solvable Groups
    DISCRETENESS PROPERTIES OF TRANSLATION NUMBERS IN SOLVABLE GROUPS GREGORY R. CONNER Abstract. We define a group to be translation proper if it carries a left- invariant metric in which the translation numbers of the non-torsion elements are nonzero and translation discrete if they are bounded away from zero. The main results of this paper are that a translation proper solvable group of finite virtual cohomological dimension is metabelian-by-finite, and that a translation discrete solvable group of finite virtual cohomological dimension, m, is a finite m extension of Z . 1. Introduction The notion of a translation number comes from Riemannian geometry. Suppose M is a compact Riemannian manifold of nonpositive sectional curvature and Mf is its universal cover, enjoying the metric induced from M. Then the covering transformation group acts as isometries on Mf. Under these conditions any covering transformation of Mf fixes a geodesic line. Since a geodesic line is isometric to R, a covering transformation must translate each point of such a fixed geodesic line by a constant amount which is called the translation number of the covering transformation. This notion can be abstracted. Let G be a group which is equipped with a metric, d, which is invariant under left multiplication by elements of G, i.e., d(xy, xz)=d(y,z) ∀ x, y, z ∈ G. We then say that d is a left-invariant group metric k k (or a metric for brevity) on G. Let kk: G −→ Z be defined by x = d(x, 1G). To fix ideas, one may consider a finitely generated group equipped with a word metric.
    [Show full text]
  • 3.4 Discrete Groups
    3.4 Discrete groups Lemma 3.4.1. Let Λ be a subgroup of R2, and let ~a;~b be two elements of Λ that are linearly independent as vectors in R2. Suppose that the paralleogram spanned by ~a and ~b contains no other elements of Λ other than ~0;~a;~b and ~a +~b. Then Λ is the group Λ = fm~a + n~b j m; n 2 Zg: (3.4) Proof. The plane is tiled up with copies of the parallellogram P spanned by the four vectors ~0;~a;~b;~a+~b. The vertices of these parallellograms are m~a+n~b for some integers m and n. Since Λ is a group all these vertices are in Λ. If Λ is not the group fm~a + n~b j m; n 2 Zg, then there exists an element x 2 Λ not in located at any of the vertices of the parallellograms. Since these parallellograms tile the whole plane there exist m and n such that x is contained in the parallellogram m~a + n~b; (m + 1)~a + n~b; m~a + (n + 1)~b and (m + 1)~a + (n + 1)~b. The element x0 = x − m~a − n~b is in the group Λ, and in the first paralleogram P . By assumption x0 is one of the four vertices of P , and it follows that also x is an vertex of the parallellograms. Hence Λ is the group fm~a + n~b j m; n 2 Zg. Definition 3.4.2. A subgroup Λ of R2 is called a lattice if there exists an > 0 such that every non-zero element ~a 2 Λ has length j~aj > .
    [Show full text]
  • Extensions of Topological and Semitopological Groups and the Product Operation
    Commentationes Mathematicae Universitatis Carolinae Aleksander V. Arhangel'skii; Miroslav Hušek Extensions of topological and semitopological groups and the product operation Commentationes Mathematicae Universitatis Carolinae, Vol. 42 (2001), No. 1, 173--186 Persistent URL: http://dml.cz/dmlcz/119232 Terms of use: © Charles University in Prague, Faculty of Mathematics and Physics, 2001 Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz Comment.Math.Univ.Carolin. 42,1 (2001)173–186 173 Extensions of topological and semitopological groups and the product operation A.V. Arhangel’skii, M. Huˇsek Abstract. The main results concern commutativity of Hewitt-Nachbin realcompactifica- tion or Dieudonn´ecompletion with products of topological groups. It is shown that for every topological group G that is not Dieudonn´ecomplete one can find a Dieudonn´e complete group H such that the Dieudonn´ecompletion of G × H is not a topological group containing G×H as a subgroup. Using Korovin’s construction of Gδ-dense orbits, we present some examples showing that some results on topological groups are not valid for semitopological groups. Keywords: topological group, Dieudonn´ecompletion, PT-group, realcompactness, Moscow space, C-embedding, product Classification: 22A05, 54H11, 54D35, 54D60 §0. Introduction Although many of our general results are valid for more general spaces, we shall assume that all the spaces under consideration are Tychonoff.
    [Show full text]