Essays on Topological Vector Spaces Quasi-Complete
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Topological Description of Riemannian Foliations with Dense Leaves
TOPOLOGICAL DESCRIPTION OF RIEMANNIAN FOLIATIONS WITH DENSE LEAVES J. A. AL¶ VAREZ LOPEZ*¶ AND A. CANDELy Contents Introduction 1 1. Local groups and local actions 2 2. Equicontinuous pseudogroups 5 3. Riemannian pseudogroups 9 4. Equicontinuous pseudogroups and Hilbert's 5th problem 10 5. A description of transitive, compactly generated, strongly equicontinuous and quasi-e®ective pseudogroups 14 6. Quasi-analyticity of pseudogroups 15 References 16 Introduction Riemannian foliations occupy an important place in geometry. An excellent survey is A. Haefliger’s Bourbaki seminar [6], and the book of P. Molino [13] is the standard reference for riemannian foliations. In one of the appendices to this book, E. Ghys proposes the problem of developing a theory of equicontinuous foliated spaces parallel- ing that of riemannian foliations; he uses the suggestive term \qualitative riemannian foliations" for such foliated spaces. In our previous paper [1], we discussed the structure of equicontinuous foliated spaces and, more generally, of equicontinuous pseudogroups of local homeomorphisms of topological spaces. This concept was di±cult to develop because of the nature of pseudogroups and the failure of having an in¯nitesimal characterization of local isome- tries, as one does have in the riemannian case. These di±culties give rise to two versions of equicontinuity: a weaker version seems to be more natural, but a stronger version is more useful to generalize topological properties of riemannian foliations. Another relevant property for this purpose is quasi-e®ectiveness, which is a generalization to pseudogroups of e®ectiveness for group actions. In the case of locally connected fo- liated spaces, quasi-e®ectiveness is equivalent to the quasi-analyticity introduced by Date: July 19, 2002. -
Dense Embeddings of Sigma-Compact, Nowhere Locallycompact Metric Spaces
proceedings of the american mathematical society Volume 95. Number 1, September 1985 DENSE EMBEDDINGS OF SIGMA-COMPACT, NOWHERE LOCALLYCOMPACT METRIC SPACES PHILIP L. BOWERS Abstract. It is proved that a connected complete separable ANR Z that satisfies the discrete n-cells property admits dense embeddings of every «-dimensional o-compact, nowhere locally compact metric space X(n e N U {0, oo}). More gener- ally, the collection of dense embeddings forms a dense Gs-subset of the collection of dense maps of X into Z. In particular, the collection of dense embeddings of an arbitrary o-compact, nowhere locally compact metric space into Hubert space forms such a dense Cs-subset. This generalizes and extends a result of Curtis [Cu,]. 0. Introduction. In [Cu,], D. W. Curtis constructs dense embeddings of a-compact, nowhere locally compact metric spaces into the separable Hilbert space l2. In this paper, we extend and generalize this result in two distinct ways: first, we show that any dense map of a a-compact, nowhere locally compact space into Hilbert space is strongly approximable by dense embeddings, and second, we extend this result to finite-dimensional analogs of Hilbert space. Specifically, we prove Theorem. Let Z be a complete separable ANR that satisfies the discrete n-cells property for some w e ÍV U {0, oo} and let X be a o-compact, nowhere locally compact metric space of dimension at most n. Then for every map f: X -» Z such that f(X) is dense in Z and for every open cover <%of Z, there exists an embedding g: X —>Z such that g(X) is dense in Z and g is °U-close to f. -
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 . -
Completeness Properties of the Generalized Compact-Open Topology on Partial Functions with Closed Domains
Topology and its Applications 110 (2001) 303–321 Completeness properties of the generalized compact-open topology on partial functions with closed domains L’ubica Holá a, László Zsilinszky b;∗ a Academy of Sciences, Institute of Mathematics, Štefánikova 49, 81473 Bratislava, Slovakia b Department of Mathematics and Computer Science, University of North Carolina at Pembroke, Pembroke, NC 28372, USA Received 4 June 1999; received in revised form 20 August 1999 Abstract The primary goal of the paper is to investigate the Baire property and weak α-favorability for the generalized compact-open topology τC on the space P of continuous partial functions f : A ! Y with a closed domain A ⊂ X. Various sufficient and necessary conditions are given. It is shown, e.g., that .P,τC/ is weakly α-favorable (and hence a Baire space), if X is a locally compact paracompact space and Y is a regular space having a completely metrizable dense subspace. As corollaries we get sufficient conditions for Baireness and weak α-favorability of the graph topology of Brandi and Ceppitelli introduced for applications in differential equations, as well as of the Fell hyperspace topology. The relationship between τC, the compact-open and Fell topologies, respectively is studied; moreover, a topological game is introduced and studied in order to facilitate the exposition of the above results. 2001 Elsevier Science B.V. All rights reserved. Keywords: Generalized compact-open topology; Fell topology; Compact-open topology; Topological games; Banach–Mazur game; Weakly α-favorable spaces; Baire spaces; Graph topology; π-base; Restriction mapping AMS classification: Primary 54C35, Secondary 54B20; 90D44; 54E52 0. -
18 | Compactification
18 | Compactification We have seen that compact Hausdorff spaces have several interesting properties that make this class of spaces especially important in topology. If we are working with a space X which is not compact we can ask if X can be embedded into some compact Hausdorff space Y . If such embedding exists we can identify X with a subspace of Y , and some arguments that work for compact Hausdorff spaces will still apply to X. This approach leads to the notion of a compactification of a space. Our goal in this chapter is to determine which spaces have compactifications. We will also show that compactifications of a given space X can be ordered, and we will look for the largest and smallest compactifications of X. 18.1 Proposition. Let X be a topological space. If there exists an embedding j : X → Y such that Y is a compact Hausdorff space then there exists an embedding j1 : X → Z such that Z is compact Hausdorff and j1(X) = Z. Proof. Assume that we have an embedding j : X → Y where Y is a compact Hausdorff space. Let j(X) be the closure of j(X) in Y . The space j(X) is compact (by Proposition 14.13) and Hausdorff, so we can take Z = j(X) and define j1 : X → Z by j1(x) = j(x) for all x ∈ X. 18.2 Definition. A space Z is a compactification of X if Z is compact Hausdorff and there exists an embedding j : X → Z such that j(X) = Z. 18.3 Corollary. -
Locally Compact Simple Rings Having Minimal Left Ideals
LOCALLY COMPACT SIMPLE RINGS HAVING MINIMAL LEFT IDEALS BY SETH WARNERC) Let A be an indiscrete locally compact primitive ring having minimal left ideals. Algebraically, A may be regarded as a dense ring of linear operators containing nonzero linear operators of finite rank on a vector space E over a division ring K, and furthermore if A is simple, then A contains only linear operators of finite rank. The topology of A induces topologies on F and A"in a natural way, so that K becomes a locally compact division ring and E a locally compact vector space over K. Theorems about locally compact division rings and locally compact vector spaces imply that if K is indiscrete, then E is necessarily finite dimensional over K, and thus A is the ring of all linear operators on a finite-dimensional vector space over a locally compact division ring. Kaplansky proved that if A has characteristic zero, then K is indeed indiscrete, but he showed by an example that E could be infinite dimensional. Kaplansky remarked [9, p. 459], however, that if A is simple, it seemed unlikely that F could be infinite dimensional, since otherwise "complete- ness appears to require the presence of linear transformations with infinite- dimensional range." Our principal result in §1 is the verification of this conjecture; however, our argument is based not on the fact that a locally compact ring is complete, but rather on the fact that a locally compact space is a Baire space and on the symmetry between minimal left and right ideals in a simple ring. -
4 Aug 2016 Metric Geometry of Locally Compact Groups
Metric geometry of locally compact groups Yves Cornulier and Pierre de la Harpe July 31, 2016 arXiv:1403.3796v4 [math.GR] 4 Aug 2016 2 Abstract This book offers to study locally compact groups from the point of view of appropriate metrics that can be defined on them, in other words to study “Infinite groups as geometric objects”, as Gromov writes it in the title of a famous article. The theme has often been restricted to finitely generated groups, but it can favourably be played for locally compact groups. The development of the theory is illustrated by numerous examples, including matrix groups with entries in the the field of real or complex numbers, or other locally compact fields such as p-adic fields, isometry groups of various metric spaces, and, last but not least, discrete group themselves. Word metrics for compactly generated groups play a major role. In the particular case of finitely generated groups, they were introduced by Dehn around 1910 in connection with the Word Problem. Some of the results exposed concern general locally compact groups, such as criteria for the existence of compatible metrics (Birkhoff-Kakutani, Kakutani-Kodaira, Struble). Other results concern special classes of groups, for example those mapping onto Z (the Bieri-Strebel splitting theorem, generalized to locally compact groups). Prior to their applications to groups, the basic notions of coarse and large-scale geometry are developed in the general framework of metric spaces. Coarse geometry is that part of geometry concerning properties of metric spaces that can be formulated in terms of large distances only. -
The Measure Algebra of a Locally Compact Hypergroup
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 179. May 1973 THE MEASURE ALGEBRA OF A LOCALLY COMPACT HYPERGROUP BY CHARLES F. DUNKL(i) ABSTRACT. A hypergroup is a locally compact space on which the space of finite regular Borel measures has a convolution structure preserving the probability measures. This paper deals only with commutative hypergroups. § 1 contains definitions, a discussion of invariant measures, and a characterization of idempotent probability measures. §2 deals with the characters of a hypergroup. §3 is about hypergroups, which have generalized translation operators (in the sense of Levitan), and subhypergroups of such. In this case the set of characters provides much information. Finally §4 discusses examples, such as the space of conjugacy classes of a compact group, certain compact homogeneous spaces, ultraspherical series, and finite hypergroups. A hypergroup is a locally compact space on which the space of finite regular Borel measures has a convolution structure preserving the probability measures. Such a structure can arise in several ways in harmonic analysis. Two major examples are furnished by the space of conjugacy classes of a compact nonabelian group, and by the two-sided cosets of certain nonnormal closed subgroups of a compact group. Another example is given by series of Jacobi polynomials. The class of hypergroups includes the class of locally compact topological semigroups. In this paper we will show that many well-known group theorems extend to the commutative hypergroup case. In §1 we discuss some basic structure and determine the idempotent probability measures. In §2 we present some elementary theory of characters of a hypergroup. -
7560 THEOREMS with PROOFS Definition a Compactification of A
7560 THEOREMS WITH PROOFS De¯nition A compacti¯cation of a completely regular space X is a compact T2- space ®X and a homeomorphic embedding ® : X ! ®X such that ®(X) is dense in ®X. Remark. To explain the separation assumptions in the above de¯nition: we will only be concerned with compacti¯cations which are Hausdor®. Note that only completely regular spaces can have a T2-compacti¯cation. Remark. A few of the concepts and results below are from ¯rst year topology, and are included here for completeness. De¯nition. Let X be a locally compact non-compact Hausdor® space. The one-point compacti¯cation !X of X is the space X [ f1g, where points of X have their usual neighborhoods, and a neighborhood of 1 has the form f1g [ (X n K) where K is a compact subset of X. In the framework of the previous de¯nition, the map ! : X ! !X is the identity map on X. Of course, X is open in its one-point compacti¯cation. We'll see that this is the case for any compacti¯cation of a locally compact X. Lemma 1. If X is a dense subset of a T2 space Z, and X is locally compact, then X is open in Z. Proof. If x 2 X, let Nx be a compact neighborhood x in X and Ux = int(Nx). Then Ux = X \ Vx for some Vx open in Z. Thus Ux is dense in Vx and so Vx ½ V x = U x ½ N x = Nx where closures are taken in Z. -
Notes on Point-Set Topology
Title. Tyrone Cutler April 29, 2020 Abstract Contents 1 Proper Maps 1 2 Local Compactness 3 3 The Compact-Open Topology 6 3.1 Further Properties of the Compact-Open Topology . 12 1 Proper Maps Proper maps make several appearances in our work. Since they can be very powerful when recognised we include a few results as background. A general reference for this section is Bourbaki [1], x1, Chapter 10. Definition 1 A continuous map f : X ! Y is said to be proper if it is closed, and if for −1 each y 2 Y the inverse image f (y) ⊆ X is compact. Lemma 1.1 If f : X ! Y is proper and K ⊆ Y is compact, then f −1(K) ⊆ X is compact. −1 Proof It will suffice to show that if fCi ⊆ f (K)gi2I is a family of closed subsets with T I Ci = ;, then there is finite subfamily whose intersection is already empty. To this end we consider the family ff(Ci) ⊆ KgI . Since f is proper this is a collection of closed subsets of K ⊆ Y whose total intersection is empty. Since K is compact, there must be finitely many of these sets, say, f(C1); : : : ; f(Cn) such that f(C1) \···\ f(Cn) = ;. But this tells us that f(C1 \···\ Cn) = ;, and the only way this can happen is if C1 \···\ Cn = ;, which was what we needed to show. Proposition 1.2 A map f : X ! Y is proper if and only if for each space Z, the map f × idZ : X × Z ! Y × Z is closed. -
University Microfilms International300 N
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LOCALLY ROELCKE PRECOMPACT POLISH GROUPS . I The
LOCALLY ROELCKE PRECOMPACT POLISH GROUPS JOSEPH ZIELINSKI Abstract. A Polish group is said to be locally Roelcke precompact if there is a neigh- borhood of the identity element that is totally bounded in the Roelcke (or lower) group uniformity. These form a subclass of the locally bounded groups, while generalizing the Roelcke precompact and locally compact Polish groups. We characterize these groups in terms of their geometric structure as those locally bounded groups whose coarsely bounded sets are all Roelcke precompact, and in terms of their uniform structure as those groups whose completions in the Roelcke uniformity are locally compact. We also assess the conditions under which this locally compact space carries the structure of a semi-topological semigroup. 1. Introduction The investigation undertaken in the present note sits at the nexus of two active areas of research into the structure of topological groups. The first of these is the topological dynamics of those groups—such as the full symmetric group on N, the unitary group of `2, or the automorphism group of the measure algebra of a standard Lebesgue space—that, while not necessarily compact or locally compact, are totally bounded in the Roelcke uniformity. This feature and the connection it engenders between such groups themselves and those spaces upon which they act have been studied extensively, for example in [1, 2, 8, 11, 18, 22, 24–26]. The second follows the discovery of C. Rosendal that topological groups carry an intrinsic large-scale geometry. An early theory of this structure can be found in the preprints [19, 20], which have been collected into and elaborated upon in the manuscript [16], and has also been explored in the papers [4, 9, 12, 27].