SOLUTIONS to EXERCISES for MATHEMATICS 205A — Part 3 III. Spaces with Special Properties
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Topology and Descriptive Set Theory
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector TOPOLOGY AND ITS APPLICATIONS ELSEVIER Topology and its Applications 58 (1994) 195-222 Topology and descriptive set theory Alexander S. Kechris ’ Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA Received 28 March 1994 Abstract This paper consists essentially of the text of a series of four lectures given by the author in the Summer Conference on General Topology and Applications, Amsterdam, August 1994. Instead of attempting to give a general survey of the interrelationships between the two subjects mentioned in the title, which would be an enormous and hopeless task, we chose to illustrate them in a specific context, that of the study of Bore1 actions of Polish groups and Bore1 equivalence relations. This is a rapidly growing area of research of much current interest, which has interesting connections not only with topology and set theory (which are emphasized here), but also to ergodic theory, group representations, operator algebras and logic (particularly model theory and recursion theory). There are four parts, corresponding roughly to each one of the lectures. The first contains a brief review of some fundamental facts from descriptive set theory. In the second we discuss Polish groups, and in the third the basic theory of their Bore1 actions. The last part concentrates on Bore1 equivalence relations. The exposition is essentially self-contained, but proofs, when included at all, are often given in the barest outline. Keywords: Polish spaces; Bore1 sets; Analytic sets; Polish groups; Bore1 actions; Bore1 equivalence relations 1. -
TOPOLOGY and ITS APPLICATIONS the Number of Complements in The
TOPOLOGY AND ITS APPLICATIONS ELSEVIER Topology and its Applications 55 (1994) 101-125 The number of complements in the lattice of topologies on a fixed set Stephen Watson Department of Mathematics, York Uniuersity, 4700 Keele Street, North York, Ont., Canada M3J IP3 (Received 3 May 1989) (Revised 14 November 1989 and 2 June 1992) Abstract In 1936, Birkhoff ordered the family of all topologies on a set by inclusion and obtained a lattice with 1 and 0. The study of this lattice ought to be a basic pursuit both in combinatorial set theory and in general topology. In this paper, we study the nature of complementation in this lattice. We say that topologies 7 and (T are complementary if and only if 7 A c = 0 and 7 V (T = 1. For simplicity, we call any topology other than the discrete and the indiscrete a proper topology. Hartmanis showed in 1958 that any proper topology on a finite set of size at least 3 has at least two complements. Gaifman showed in 1961 that any proper topology on a countable set has at least two complements. In 1965, Steiner showed that any topology has a complement. The question of the number of distinct complements a topology on a set must possess was first raised by Berri in 1964 who asked if every proper topology on an infinite set must have at least two complements. In 1969, Schnare showed that any proper topology on a set of infinite cardinality K has at least K distinct complements and at most 2” many distinct complements. -
Notes on Topology
Notes on Topology Notes on Topology These are links to PostScript files containing notes for various topics in topology. I learned topology at M.I.T. from Topology: A First Course by James Munkres. At the time, the first edition was just coming out; I still have the photocopies we were given before the printed version was ready! Hence, I'm a bit biased: I still think Munkres' book is the best book to learn from. The writing is clear and lively, the choice of topics is still pretty good, and the exercises are wonderful. Munkres also has a gift for naming things in useful ways (The Pasting Lemma, the Sequence Lemma, the Tube Lemma). (I like Paul Halmos's suggestion that things be named in descriptive ways. On the other hand, some names in topology are terrible --- "first countable" and "second countable" come to mind. They are almost as bad as "regions of type I" and "regions of type II" which you still sometimes encounter in books on multivariable calculus.) I've used Munkres both of the times I've taught topology, the most recent occasion being this summer (1999). The lecture notes below follow the order of the topics in the book, with a few minor variations. Here are some areas in which I decided to do things differently from Munkres: ● I prefer to motivate continuity by recalling the epsilon-delta definition which students see in analysis (or calculus); therefore, I took the pointwise definition as a starting point, and derived the inverse image version later. ● I stated the results on quotient maps so as to emphasize the universal property. -
One Dimensional Locally Connected S-Spaces
One Dimensional Locally Connected S-spaces ∗ Joan E. Hart† and Kenneth Kunen‡§ October 28, 2018 Abstract We construct, assuming Jensen’s principle ♦, a one-dimensional locally connected hereditarily separable continuum without convergent sequences. 1 Introduction All topologies discussed in this paper are assumed to be Hausdorff. A continuum is any compact connected space. A nontrivial convergent sequence is a convergent ω–sequence of distinct points. As usual, dim(X) is the covering dimension of X; for details, see Engelking [5]. “HS” abbreviates “hereditarily separable”. We shall prove: Theorem 1.1 Assuming ♦, there is a locally connected HS continuum Z such that dim(Z)=1 and Z has no nontrivial convergent sequences. Note that points in Z must have uncountable character, so that Z is not heredi- tarily Lindel¨of; thus, Z is an S-space. arXiv:0710.1085v1 [math.GN] 4 Oct 2007 Spaces with some of these features are well-known from the literature. A compact F-space has no nontrivial convergent sequences. Such a space can be a continuum; for example, the Cechˇ remainder β[0, 1)\[0, 1) is connected, although not locally con- nected; more generally, no infinite compact F-space can be either locally connected or HS. In [13], van Mill constructs, under the Continuum Hypothesis, a locally con- nected continuum with no nontrivial convergent sequences. Van Mill’s example, ∗2000 Mathematics Subject Classification: Primary 54D05, 54D65. Key Words and Phrases: one-dimensional, Peano continuum, locally connected, convergent sequence, Menger curve, S-space. †University of Wisconsin, Oshkosh, WI 54901, U.S.A., [email protected] ‡University of Wisconsin, Madison, WI 53706, U.S.A., [email protected] §Both authors partially supported by NSF Grant DMS-0456653. -
Math 501. Homework 2 September 15, 2009 ¶ 1. Prove Or
Math 501. Homework 2 September 15, 2009 ¶ 1. Prove or provide a counterexample: (a) If A ⊂ B, then A ⊂ B. (b) A ∪ B = A ∪ B (c) A ∩ B = A ∩ B S S (d) i∈I Ai = i∈I Ai T T (e) i∈I Ai = i∈I Ai Solution. (a) True. (b) True. Let x be in A ∪ B. Then x is in A or in B, or in both. If x is in A, then B(x, r) ∩ A , ∅ for all r > 0, and so B(x, r) ∩ (A ∪ B) , ∅ for all r > 0; that is, x is in A ∪ B. For the reverse containment, note that A ∪ B is a closed set that contains A ∪ B, there fore, it must contain the closure of A ∪ B. (c) False. Take A = (0, 1), B = (1, 2) in R. (d) False. Take An = [1/n, 1 − 1/n], n = 2, 3, ··· , in R. (e) False. ¶ 2. Let Y be a subset of a metric space X. Prove that for any subset S of Y, the closure of S in Y coincides with S ∩ Y, where S is the closure of S in X. ¶ 3. (a) If x1, x2, x3, ··· and y1, y2, y3, ··· both converge to x, then the sequence x1, y1, x2, y2, x3, y3, ··· also converges to x. (b) If x1, x2, x3, ··· converges to x and σ : N → N is a bijection, then xσ(1), xσ(2), xσ(3), ··· also converges to x. (c) If {xn} is a sequence that does not converge to y, then there is an open ball B(y, r) and a subsequence of {xn} outside B(y, r). -
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.......................... -
Isolated Points, Duality and Residues
JOURNAL OF PURE AND APPLIED ALGEBRA Journal of Pure and Applied Algebra 117 % 118 (1997) 469-493 Isolated points, duality and residues B . Mourrain* INDIA, Projet SAFIR, 2004 routes des Lucioles, BP 93, 06902 Sophia-Antipolir, France This work is dedicated to the memory of J. Morgenstem. Abstract In this paper, we are interested in the use of duality in effective computations on polynomials. We represent the elements of the dual of the algebra R of polynomials over the field K as formal series E K[[a]] in differential operators. We use the correspondence between ideals of R and vector spaces of K[[a]], stable by derivation and closed for the (a)-adic topology, in order to construct the local inverse system of an isolated point. We propose an algorithm, which computes the orthogonal D of the primary component of this isolated point, by integration of polynomials in the dual space K[a], with good complexity bounds. Then we apply this algorithm to the computation of local residues, the analysis of real branches of a locally complete intersection curve, the computation of resultants of homogeneous polynomials. 0 1997 Elsevier Science B.V. 1991 Math. Subj. Class.: 14B10, 14Q20, 15A99 1. Introduction Considering polynomials as algorithms (that compute a value at a given point) in- stead of lists of terms, has lead to recent interesting developments, both from a practical and a theoretical point of view. In these approaches, evaluations at points play an im- portant role. We would like to pursue this idea further and to study evaluations by themselves. -
Math 3283W: Solutions to Skills Problems Due 11/6 (13.7(Abc)) All
Math 3283W: solutions to skills problems due 11/6 (13.7(abc)) All three statements can be proven false using counterexamples that you studied on last week's skills homework. 1 (a) Let S = f n jn 2 Ng. Every point of S is an isolated point, since given 1 1 1 n 2 S, there exists a neighborhood N( n ; ") of n that contains no point of S 1 1 1 different from n itself. (We could, for example, take " = n − n+1 .) Thus the set P of isolated points of S is just S. But P = S is not closed, since 0 is a boundary point of S which nevertheless does not belong to S (every neighbor- 1 hood of 0 contains not only 0 but, by the Archimedean property, some n , thus intersects both S and its complement); if S were closed, it would contain all of its boundary points. (b) Let S = N. Then (as you saw on the last skills homework) int(S) = ;. But ; is closed, so its closure is itself; that is, cl(int(S)) = cl(;) = ;, which is certainly not the same thing as S. (c) Let S = R n N. The closure of S is all of R, since every natural number n is a boundary point of R n N and thus belongs to its closure. But R is open, so its interior is itself; that is, int(cl(S)) = int(R) = R, which is certainly not the same thing as S. (13.10) Suppose that x is an isolated point of S, and let N = (x − "; x + ") (for some " > 0) be an arbitrary neighborhood of x. -
Polish Spaces and Baire Spaces
Polish spaces and Baire spaces Jordan Bell [email protected] Department of Mathematics, University of Toronto June 27, 2014 1 Introduction These notes consist of me working through those parts of the first chapter of Alexander S. Kechris, Classical Descriptive Set Theory, that I think are impor- tant in analysis. Denote by N the set of positive integers. I do not talk about universal spaces like the Cantor space 2N, the Baire space NN, and the Hilbert cube [0; 1]N, or \localization", or about Polish groups. If (X; τ) is a topological space, the Borel σ-algebra of X, denoted by BX , is the smallest σ-algebra of subsets of X that contains τ. BX contains τ, and is closed under complements and countable unions, and rather than talking merely about Borel sets (elements of the Borel σ-algebra), we can be more specific by talking about open sets, closed sets, and sets that are obtained by taking countable unions and complements. Definition 1. An Fσ set is a countable union of closed sets. A Gδ set is a complement of an Fσ set. Equivalently, it is a countable intersection of open sets. If (X; d) is a metric space, the topology induced by the metric d is the topology generated by the collection of open balls. If (X; τ) is a topological space, a metric d on the set X is said to be compatible with τ if τ is the topology induced by d.A metrizable space is a topological space whose topology is induced by some metric, and a completely metrizable space is a topological space whose topology is induced by some complete metric. -
Two Characterizations of Linear Baire Spaces 205
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 45, Number 2, August 1974 TWOCHARACTERIZATIONS OF LINEAR BAIRE SPACES STEPHEN A. SAXON1 ABSTRACT. The Wilansky-Klee conjecture is equivalent to the (un- proved) conjecture that every dense, 1-codimensional subspace of an arbitrary Banach space is a Baire space (second category in itself). The following two characterizations may be useful in dealing with this conjecture: (i) A topological vector space is a Baire space if and on- ly if every absorbing, balanced, closed set is a neighborhood of some point, (ii) A topological vector space is a Baire space if and only if it cannot be covered by countably many nowhere dense sets, each of which is a union of lines (1-dimensional subspaces). Characterization (i) has a more succinct form, using the definition of Wilansky's text [8, p. 224]: a topological vector space is a Baire space if and only if it has the t property. Introduction. The Wilansky-Klee conjecture (see [3], [7]) is equivalent to the conjecture that every dense, 1-codimensional subspace of a Banach space is a Baire space. In [4], [5], [6], [7] it is shown that every counta- ble-codimensional subspace of a locally convex space which is "nearly" a Baire space is, itself, "nearly" a Baire space. (Theorem 1 of this paper indicates how "nearly Baire" a barrelled space is: Wilansky's class of W'barrelled spaces [9,p. 44] is precisely the class of linear Baire spaces.) The theorem [7] that every countable-codimensional subspace of an unor- dered Baire-like space is unordered Baire-like is the closest to an affirmation of the conjecture. -
TOPOLOGY PRELIM REVIEW 2021: LIST THREE Topic 1: Baire Property and Gδ Sets. Definition. X Is a Baire Space If a Countable Inte
TOPOLOGY PRELIM REVIEW 2021: LIST THREE Topic 1: Baire property and Gδ sets. Definition. X is a Baire space if a countable intersection of open, dense subsets of X is dense in X. Complete metric spaces and locally compact spaces are Baire spaces. Def. X is locally compact if for all x 2 X, and all open Ux, there exists Vx with compact closure V x ⊂ Ux. 1. X is locally compact , for all C ⊂ X compact, and all open U ⊃ C, there exists V open with compact closure, so that: C ⊂ V ⊂ V ⊂ U. 2. Def: A set E ⊂ X is nowhere dense in X if its closure E has empty interior. Show: X is a Baire space , any countable union of nowhere dense sets has empty interior. 3. A complete metric space without isolated points is uncountable. (Hint: Baire property, complements of one-point sets.) 4. Uniform boundedness principle. X complete metric, F ⊂ C(X) a family of continuous functions, bounded at each point: (8a 2 X)(9M(a) > 0)(8f 2 F)jf(a)j ≤ M(a): Then there exists a nonempty open set U ⊂ X so that F is eq¨uibounded over U{there exists a constant C > 0 so that: (8f 2 F)(8x 2 U)jf(x)j ≤ C: Hint: For n ≥ 1, consider An = fx 2 X; jf(x)j ≤ n; 8f 2 Fg. Use Baire's property. An important application of 4. is the uniform boundedness principle for families of bounded linear operators F ⊂ L(E; F ), where E; F are Banach spaces. -
Solutions to Exercises in Munkres
April 21, 2006 Munkres §29 Ex. 29.1. Closed intervals [a, b] ∩ Q in Q are not compact for they are not even sequentially compact [Thm 28.2]. It follows that all compact subsets of Q have empty interior (are nowhere dense) so Q can not be locally compact. To see that compact subsets of Q are nowhere dense we may argue as follows: If C ⊂ Q is compact and C has an interior point then there is a whole open interval (a, b) ∩ Q ⊂ C and also [a, b] ∩ Q ⊂ C for C is closed (as a compact subset of a Hausdorff space [Thm 26.3]). The closed subspace [a, b] ∩ Q of C is compact [Thm 26.2]. This contradicts that no closed intervals of Q are compact. Ex. 29.2. Q (a). Assume that the product Xα is locally compact. Projections are continuous and open [Ex 16.4], so Xα is locally compact for all α [Ex 29.3]. Furthermore, there are subspaces U ⊂ C such that U is nonempty and open and C is compact. Since πα(U) = Xα for all but finitely many α, also πα(C) = Xα for all but finitely many α. But C is compact so also πα(C) is compact. Q (b). We have Xα = X1 × X2 where X1 is a finite product of locally compact spaces and X2 is a product of compact spaces. It is clear that finite products of locally compact spaces are locally compact for finite products of open sets are open and all products of compact spaces are compact by Tychonoff.