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ELEMENTARY TOPOLOGY I 1. Introduction 3 2. Metric Spaces 4 2.1
ELEMENTARY TOPOLOGY I ALEX GONZALEZ 1. Introduction3 2. Metric spaces4 2.1. Continuous functions8 2.2. Limits 10 2.3. Open subsets and closed subsets 12 2.4. Continuity, convergence, and open/closed subsets 15 3. Topological spaces 18 3.1. Interior, closure and boundary 20 3.2. Continuous functions 23 3.3. Basis of a topology 24 3.4. The subspace topology 26 3.5. The product topology 27 3.6. The quotient topology 31 3.7. Other constructions 33 4. Connectedness 35 4.1. Path-connected spaces 40 4.2. Connected components 41 5. Compactness 43 5.1. Compact subspaces of Rn 46 5.2. Nets and Tychonoff’s Theorem 48 6. Countability and separation axioms 52 6.1. The countability axioms 52 6.2. The separation axioms 54 7. Urysohn metrization theorem 61 8. Topological manifolds 65 8.1. Compact manifolds 66 REFERENCES 67 Contents 1 2 ALEX GONZALEZ ELEMENTARY TOPOLOGY I 3 1. Introduction When we consider properties of a “reasonable” function, probably the first thing that comes to mind is that it exhibits continuity: the behavior of the function at a certain point is similar to the behavior of the function in a small neighborhood of the point. What’s more, the composition of two continuous functions is also continuous. Usually, when we think of a continuous functions, the first examples that come to mind are maps f : R ! R: • the identity function, f(x) = x for all x 2 R; • a constant function f(x) = k; • polynomial functions, for instance f(x) = xn, for some n 2 N; • the exponential function g(x) = ex; • trigonometric functions, for instance h(x) = cos(x). -
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.......................... -
Lecture 13: Basis for a Topology
Lecture 13: Basis for a Topology 1 Basis for a Topology Lemma 1.1. Let (X; T) be a topological space. Suppose that C is a collection of open sets of X such that for each open set U of X and each x in U, there is an element C 2 C such that x 2 C ⊂ U. Then C is the basis for the topology of X. Proof. In order to show that C is a basis, need to show that C satisfies the two properties of basis. To show the first property, let x be an element of the open set X. Now, since X is open, then, by hypothesis there exists an element C of C such that x 2 C ⊂ X. Thus C satisfies the first property of basis. To show the second property of basis, let x 2 X and C1;C2 be open sets in C such that x 2 C1 and x 2 C2. This implies that C1 \ C2 is also an open set in C and x 2 C1 \ C2. Then, by hypothesis, there exists an open set C3 2 C such that x 2 C3 ⊂ C1 \ C2. Thus, C satisfies the second property of basis too and hence, is indeed a basis for the topology on X. On many occasions it is much easier to show results about a topological space by arguing in terms of its basis. For example, to determine whether one topology is finer than the other, it is easier to compare the two topologies in terms of their bases. -
A Discussion on Analytical Study of Semi-Closed Set in Topological Space
The International journal of analytical and experimental modal analysis ISSN NO:0886-9367 A discussion on analytical study of Semi-closed set in topological space 1 Dr. Priti Kumari, 2 Sukesh Kumar Das, 3 Dr. Ranjana & 4 Rupesh Kumar 1 & 2 Guest Assistant Professor, Department of Mathematics Saharsa College of Engineering , Saharsa ( 852201 ), Bihar, INDIA 3 University Professor, University department of Mathematics Tilka Manjhi Bhagalpur University, Bhagalpur ( 812007 ), Bihar , INDIA 4 M. Sc., Department of Physics A. N. College Patna, Univ. of Patna ( 800013 ), Bihar, INDIA [email protected] , [email protected] , [email protected] & [email protected] Abstract : In this paper, we introduce a new class of sets in the topological space, namely Semi- closed sets in the topological space. We find characterizations of these sets. Further, we study some fundamental properties of Semi-closed sets in the topological space. Keywords : Open set, Closed set, Interior of a set & Closure of a set. I. Introduction The term Semi-closed set which is a weak form of closed set in a topological space and it is introduced and defined by the mathematician N. Biswas [10] in the year 1969. The term Semi- closure of a set in a topological space defined and introduced by two mathematician Crossley S. G. & Hildebrand S. K. [3,4] in the year 1971. The mathematician N. Levine [1] also defined and studied the term generalized closed sets in the topological space in Jan 1970. The term Semi- Interior point & Semi-Limit point of a subset of a topological space was defined and studied by the mathematician P. -
Advance Topics in Topology - Point-Set
ADVANCE TOPICS IN TOPOLOGY - POINT-SET NOTES COMPILED BY KATO LA 19 January 2012 Background Intervals: pa; bq “ tx P R | a ă x ă bu ÓÓ , / / calc. notation set theory notation / / \Open" intervals / pa; 8q ./ / / p´8; bq / / / -/ ra; bs; ra; 8q: Closed pa; bs; ra; bq: Half-openzHalf-closed Open Sets: Includes all open intervals and union of open intervals. i.e., p0; 1q Y p3; 4q. Definition: A set A of real numbers is open if @ x P A; D an open interval contain- ing x which is a subset of A. Question: Is Q, the set of all rational numbers, an open set of R? 1 1 1 - No. Consider . No interval of the form ´ "; ` " is a subset of . We can 2 2 2 Q 2 ˆ ˙ ask a similar question in R . 2 Is R open in R ?- No, because any disk around any point in R will have points above and below that point of R. Date: Spring 2012. 1 2 NOTES COMPILED BY KATO LA Definition: A set is called closed if its complement is open. In R, p0; 1q is open and p´8; 0s Y r1; 8q is closed. R is open, thus Ø is closed. r0; 1q is not open or closed. In R, the set t0u is closed: its complement is p´8; 0q Y p0; 8q. In 2 R , is tp0; 0qu closed? - Yes. Chapter 2 - Topological Spaces & Continuous Functions Definition:A topology on a set X is a collection T of subsets of X satisfying: (1) Ø;X P T (2) The union of any number of sets in T is again, in the collection (3) The intersection of any finite number of sets in T , is again in T Alternative Definition: ¨ ¨ ¨ is a collection T of subsets of X such that Ø;X P T and T is closed under arbitrary unions and finite intersections. -
POINT SET TOPOLOGY Definition 1 a Topological Structure On
POINT SET TOPOLOGY De¯nition 1 A topological structure on a set X is a family (X) called open sets and satisfying O ½ P (O ) is closed for arbitrary unions 1 O (O ) is closed for ¯nite intersections. 2 O De¯nition 2 A set with a topological structure is a topological space (X; ) O ; = 2;Ei = x : x Eifor some i = [ [i f 2 2 ;g ; so is always open by (O ) ; 1 ; = 2;Ei = x : x Eifor all i = X \ \i f 2 2 ;g so X is always open by (O2). Examples (i) = (X) the discrete topology. O P (ii) ; X the indiscrete of trivial topology. Of; g These coincide when X has one point. (iii) =the rational line. Q =set of unions of open rational intervals O De¯nition 3 Topological spaces X and X 0 are homomorphic if there is an isomorphism of their topological structures i.e. if there is a bijection (1-1 onto map) of X and X 0 which generates a bijection of and . O O e.g. If X and X are discrete spaces a bijection is a homomorphism. (see also Kelley p102 H). De¯nition 4 A base for a topological structure is a family such that B ½ O every o can be expressed as a union of sets of 2 O B Examples (i) for the discrete topological structure x x2X is a base. f g (ii) for the indiscrete topological structure ; X is a base. f; g (iii) For , topologised as before, the set of bounded open intervals is a base.Q 1 (iv) Let X = 0; 1; 2 f g Let = (0; 1); (1; 2); (0; 12) . -
Arxiv:1910.07913V4 [Math.LO]
REPRESENTATIONS AND THE FOUNDATIONS OF MATHEMATICS SAM SANDERS Abstract. The representation of mathematical objects in terms of (more) ba- sic ones is part and parcel of (the foundations of) mathematics. In the usual foundations of mathematics, i.e. ZFC set theory, all mathematical objects are represented by sets, while ordinary, i.e. non-set theoretic, mathematics is rep- resented in the more parsimonious language of second-order arithmetic. This paper deals with the latter representation for the rather basic case of continu- ous functions on the reals and Baire space. We show that the logical strength of basic theorems named after Tietze, Heine, and Weierstrass, changes signifi- cantly upon the replacement of ‘second-order representations’ by ‘third-order functions’. We discuss the implications and connections to the Reverse Math- ematics program and its foundational claims regarding predicativist mathe- matics and Hilbert’s program for the foundations of mathematics. Finally, we identify the problem caused by representations of continuous functions and formulate a criterion to avoid problematic codings within the bigger picture of representations. 1. Introduction Lest we be misunderstood, let our first order of business be to formulate the following blanket caveat: any formalisation of mathematics generally involves some kind of representation (aka coding) of mathematical objects in terms of others. Now, the goal of this paper is to critically examine the role of representations based on the language of second-order arithmetic; such an examination perhaps unsurpris- ingly involves the comparison of theorems based on second-order representations versus theorems formulated in third-order arithmetic. To be absolutely clear, we do not claim that the latter represent the ultimate mathematical truth, nor do we arXiv:1910.07913v5 [math.LO] 9 Aug 2021 (wish to) downplay the role of representations in third-order arithmetic. -
Some Examples and Counterexamples of Advanced Compactness in Topology
International Journal of Research in Engineering and Science (IJRES) ISSN (Online): 2320-9364, ISSN (Print): 2320-9356 www.ijres.org Volume 9 Issue 1 ǁ 2021 ǁ PP. 10-16 Some Examples and Counterexamples of Advanced Compactness in Topology Pankaj Goswami Department of Mathematics, University of Gour Banga, Malda, 732102, West Bengal, India ABSTRACT: The examples and counter examples are always usefull for better comprehension of underlying concept in a theorem or definition .Compactness has come to be one of the most importent and useful topic in advanced mathematics.This paper is an attempt to fill in some of the information that the standard textbook treatment of compactness leaves out, and giving some constructive significative counterexamples of Advanced Compactness in Topology . KEYWORDS: open cover, compact, connected, topological space, example, counterexamples, continuous function, locally compact, countably compact, limit point compact, sequentially compact, lindelöf , paracompact --------------------------------------------------------------------------------------------------------------------------------------- Date of Submission: 15-01-2021 Date of acceptance: 30-01-2021 --------------------------------------------------------------------------------------------------------------------------------------- I. INTRODUCTION We begin by presenting some definitions, notations,examples and some theorems that are essential for concept of compactness in advanced topology .For more detailed, see [1-2], [7-10] and others, in references .If X is a closed bounded subset of the real line ℝ, then any family of open sets in ℝ whose union contains X has a finite sub family whose union also contains X . If X is a metric space or topological space in its own right , then the above proposition can be thought as saying that any class of open sets in X whose union is equal to X has a finite subclass whose union is also equal to X. -
16. Compactness
16. Compactness 1 Motivation While metrizability is the analyst's favourite topological property, compactness is surely the topologist's favourite topological property. Metric spaces have many nice properties, like being first countable, very separative, and so on, but compact spaces facilitate easy proofs. They allow you to do all the proofs you wished you could do, but never could. The definition of compactness, which we will see shortly, is quite innocuous looking. What compactness does for us is allow us to turn infinite collections of open sets into finite collections of open sets that do essentially the same thing. Compact spaces can be very large, as we will see in the next section, but in a strong sense every compact space acts like a finite space. This behaviour allows us to do a lot of hands-on, constructive proofs in compact spaces. For example, we can often take maxima and minima where in a non-compact space we would have to take suprema and infima. We will be able to intersect \all the open sets" in certain situations and end up with an open set, because finitely many open sets capture all the information in the whole collection. We will specifically prove an important result from analysis called the Heine-Borel theorem n that characterizes the compact subsets of R . This result is so fundamental to early analysis courses that it is often given as the definition of compactness in that context. 2 Basic definitions and examples Compactness is defined in terms of open covers, which we have talked about before in the context of bases but which we formally define here. -
14. Arbitrary Products
14. Arbitrary products Contents 1 Motivation 2 2 Background and preliminary definitions2 N 3 The space Rbox 4 N 4 The space Rprod 6 5 The uniform topology on RN, and completeness 12 6 Summary of results about RN 14 7 Arbitrary products 14 8 A final note on completeness, and completions 19 c 2018{ Ivan Khatchatourian 1 14. Arbitrary products 14.2. Background and preliminary definitions 1 Motivation In section 8 of the lecture notes we discussed a way of defining a topology on a finite product of topological spaces. That definition more or less agreed with our intuition. We should expect products of open sets in each coordinate space to be open in the product, for example, and the only issue that arises is that these sets only form a basis rather than a topology. So we simply generate a topology from them and call it the product topology. This product topology \played nicely" with the topologies on each of the factors in a number of ways. In that earlier section, we also saw that every topological property we had studied up to that point other than ccc-ness was finitely productive. Since then we have developed some new topological properties like regularity, normality, and metrizability, and learned that except for normality these are also finitely productive. So, ultimately, most of the properties we have studied are finitely productive. More importantly, the proofs that they are finitely productive have been easy. Look back for example to your proof that separability is finitely productive, and you will see that there was almost nothing to do. -
URYSOHN's THEOREM and TIETZE EXTENSION THEOREM Definition
URYSOHN'S THEOREM AND TIETZE EXTENSION THEOREM Tianlin Liu [email protected] Mathematics Department Jacobs University Bremen Campus Ring 6, 28759, Bremen, Germany Definition 0.1. Let x; y topological space X. We define the following properties of topological space X: ∈ T0: If x y, there is an open set containing x but not y or an open set containing y but not x. ≠ T1: If x y, there is an open set containing y but not x. T2: If x≠ y, there are disjoint open sets U; V with x U and y V . ≠ ∈ T3∈: X is a T1 space, and for any closed set A X and any x AC there are disjoint open sets U; V with x U and A V . ⊂ T4∈: X is a T1 space, and for any disjoint closed∈ sets A, ⊂B in X there are disjoint open sets U, V with A Uand B V . We say a T2 space is a Hausdorff space, a T⊂3 space is⊂ a regular space, a T4 space is a normal space. Definition 0.2. C X; a; b Space of all continuous a; b valued functions on X. ( [ ]) ∶= [ ] Date: February 11, 2016. 1 URYSOHN'S THEOREM AND TIETZE EXTENSION THEOREM 2 Theorem 0.3. (Urysohn's Lemma) Let X be a normal space. If A and B are disjoint closed sets in X, there exists f C X; 0; 1 such that f 0 on A and f 1 on B. Proof. ∈ ( [ ]) = = Step 1: Define a large collection of open sets in X (Lemma 4.14 in [1]) Let D be the set of dyadic rationals in 0; 1 , that is, D 1 1 3 1 3 7 1; 0; 2 ; 4 ; 4 ; 8 ; 8 ; 8 ::: . -
Some Properties of Θ-Open Sets
Divulgaciones Matem¶aticasVol. 12 No. 2(2004), pp. 161{169 Some Properties of θ-open Sets Algunas Propiedades de los Conjuntos θ-abiertos M. Caldas ([email protected]) Departamento de Matematica Aplicada, Universidade Federal Fluminense, Rua Mario Santos Braga, s/n 24020-140, Niteroi, RJ Brasil. S. Jafari ([email protected]) Department of Mathematics and Physics, Roskilde University, Postbox 260, 4000 Roskilde, Denmark. M. M. Kov¶ar([email protected]) Department of Mathematics, Faculty of Electrical Engineering and Computer Sciences Technical University of Brno, Technick ¶a8 616 69 Brno, Czech Republic. Abstract In the present paper, we introduce and study topological properties of θ-derived, θ-border, θ-frontier and θ-exterior of a set using the con- cept of θ-open sets and study also other properties of the well known notions of θ-closure and θ-interior. Key words and phrases: θ-open, θ-closure, θ-interior, θ-border, θ- frontier, θ-exterior. Resumen En el presente ert¶³culose introducen y estudian las propiedades to- pol¶ogicasdel θ-derivedo, θ-borde, θ-frontera y θ-exterior de un conjunto usando el concepto de conjunto θ-abierto y estudiando tambi¶enotras propiedades de las nociones bien conocidas de θ-clausura y θ-interior. Palabras y frases clave: θ-abierto, θ-clausura, θ-interior, θ-borde, θ-frontera, θ-exterior. Received 2003/09/30. Revised 2004/10/15. Accepted 2004/10/19. MSC (2000): Primary 54A20, 54A05. 162 M. Caldas, S. Jafari, M. M. Kov¶ar 1 Introduction The notions of θ-open subsets, θ-closed subsets and θ-closure where introduced by Veli·cko [14] for the purpose of studying the important class of H-closed spaces in terms of arbitrary ¯berbases.