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metric space topology pdf

The ﬁrst goal of this course is then to deﬁne metric spaces and continuous functions between metric spaces. The following are equivalent: (i) A and B are mutually separated. 1 Distance A metric space can be thought of as a very basic space having a geometry, with only a few axioms. A Theorem of Volterra Vito 15 9. Year: 2005. h�b```� ���@(�����с$���!��FG�N�D�o��
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�LF�S�D5 The function f is called continuous if, for all x 0 2 X , it is continuous at x 0. General Topology 1 Metric and topological spaces The deadline for handing this work in is 1pm on Monday 29 September 2014. De nition and basic properties 79 8.2. x, then x is the only accumulation point of fxng1 n 1 Proof. endobj
(Alternative characterization of the closure). The metric space (í µí± , í µí± ) is denoted by í µí² [í µí± , í µí± ]. Topology Generated by a Basis 4 4.1. Polish Space. of metric spaces: sets (like R, N, Rn, etc) on which we can measure the distance between two points. 3 Topology of Metric Spaces We use standard notions from the constructive theory of metric spaces [2, 20]. If is closed, then . To this end, the book boasts of a lot of pictures. 4.2 Theorem. Details of where to hand in, how the work will be assessed, etc., can be found in the FAQ on the course Learn page. Every metric space (X;d) has a topology which is induced by its metric. �F�%���K� ��X,�h#�F��r'e?r��^o�.re�f�cTbx°�1/�� ?����~oGiOc!�m����� ��(��t�� :���e`�g������vd`����3& }��]
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Topology of metric space Metric Spaces Page 3 . The open ball around xof radius ", or more brie Metric Space Topology Open sets. Continuous Functions 12 8.1. Metric spaces. For a metric space (X;d) the (metric) ball centered at x2Xwith radius r>0 is the set B(x;r) = fy2Xjd(x;y) /ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 594.6 843.24] /Contents 4 0 R/Group<>/Tabs/S/StructParents 1>>
De nition { Metrisable space A topological space (X;T) is called metrisable, if there exists a metric on Xsuch that the topology Tis induced by this metric. Please take care over communication and presentation. iff is closed. A metric space is a set X where we have a notion of distance. of topology will also give us a more generalized notion of the meaning of open and closed sets. In fact, a "metric" is the generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance. (iii) A and B are both closed sets. The same set can be given diﬀerent ways of measuring distances. Strange as it may seem, the set R2 (the plane) is one of these sets. then B is called a base for the topology τ. Let ϵ>0 be given. 3.1 Euclidean n-space The set Un is an extension of the concept of the Cartesian product of two sets that was studied in MAT108. This is a text in elementary real analysis. Applications 82 9. 4 0 obj
You learn about properties of these spaces and generalise theorems like IVT and EVT which you learnt from real analysis. h�bbd```b``� ";@$���D Real Variables with Basic Metric Space Topology. Open, closed and compact sets . 1 0 obj
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Prof. Corinna Ulcigrai Metric Spaces and Topology 1.1 Metric Spaces and Basic Topology notions In this section we brie y overview some basic notions about metric spaces and topology. Free download PDF Best Topology And Metric Space Hand Written Note. We shall define intuitive topological definitions through it (that will later be converted to the real topological definition), and convert (again, intuitively) calculus definitions of properties (like convergence and continuity) to their topological definition. Quotient spaces 52 6.1. Basic concepts Topology … In nitude of Prime Numbers 6 5. First part of this course note presents a rapid overview of metric spaces to set the scene for the main topic of topological spaces.Further it covers metric spaces, Continuity and open sets for metric spaces, Closed sets for metric spaces, Topological spaces, Interior and closure, More on topological structures, Hausdorff spaces and Compactness. Definition: If X is a topological space and FX⊂ , then F is said to be closed if FXFc = ∼ is open. Topics covered includes: Upper and Lower Limits of Sequences of Real Numbers, Continuous Functions, Differentiation, Riemann-Stieltjes Integration, Unifom Convergence … In precise mathematical notation, one has (8 x 0 2 X )(8 > 0)( 9 > 0) (8 x 2 f x 0 2 X jd X (x 0;x 0) < g); d Y (f (x 0);f (x )) < : Denition 2.1.25. 'a ]��i�U8�"Tt�L�KS���+[x�. 1.1 Metric Spaces Deﬁnition 1.1.1. Topology of Metric Spaces S. Kumaresan. 2 0 obj
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Quotient topology 52 6.2. By the deﬁnition of convergence, 9N such that d„xn;x” <ϵ for all n N. fn 2 N: n Ng is inﬁnite, so x is an accumulation point. PDF | We define the concepts of -metric in sets over -complete Boolean algebra and obtain some applications of them on the theory of topology. Since Yet another characterization of closure. The purpose of this chapter is to introduce metric spaces and give some deﬁnitions and examples. � ��
Ra���"y��ȸ}@����.�b*��`�L����d(H}�)[u3�z�H�3y�����8��QD To encourage the geometric thinking, I have chosen large number of examples which allow us to draw pictures and develop our intuition and draw conclusions, generate ideas for proofs. �)@ Metric spaces are generalizations of the real line, in which some of the theorems that hold for R remain valid. (ii) A and B are both open sets. NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. Proof. Proof. Subspace Topology 7 7. endobj
10 CHAPTER 9. �fWx��~ The particular distance function must satisfy the following conditions: ��So�goir����(�ZV����={�Y8�V��|�2>uC�C>�����e�N���gz�>|�E�:��8�V,��9ڼ淺mgoe��Q&]�]�ʤ� .$�^��-�=w�5q����%�ܕv���drS�J��� 0I In mathematics, a metric space is a set for which distances between all members of the set are defined. Product Topology 6 6. But if we wish, for example, to classify surfaces or knots, we want to think of the objects as rubbery. Mn�qn�:�֤���u6�
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p��i�x�A�r�ѵTZ��X��i��Y����D�a��9�9�A�p�����3��0>�A.;o;�X��7U9�x��. That is, if x,y ∈ X, then d(x,y) is the “distance” between x and y. METRIC SPACES, TOPOLOGY, AND CONTINUITY Lemma 1.1. When we discuss probability theory of random processes, the underlying sample spaces and σ-ﬁeld structures become quite complex. Notes: 1. We want to endow this set with a metric; i.e a way to measure distances between elements of X.A distanceor metric is a function d: X×X →R such that if we take two elements x,y∈Xthe number d(x,y) gives us the distance between them. Closed Sets, Hausdor Spaces, and Closure of a Set 9 8. Notes on Metric Spaces These notes introduce the concept of a metric space, which will be an essential notion throughout this course and in others that follow. Metric spaces and topology. Topological Spaces 3 3. A metric on a space induces topological properties like open and closed sets, which lead to the study of more abstract topological spaces. The open ball is the building block of metric space topology. Fix then Take . set topology has its main value as a language for doing ‘continuous geome- try’; I believe it is important that the subject be presented to the student in this way, rather than as a … 2 2. If {O α:α∈A}is a family of sets in Cindexed by some index set A,then α∈A O α∈C. endobj
For define Then iff Remark. Skorohod metric and Skorohod space. Categories: Mathematics\\Geometry and Topology. A subset S of the set X is open in the metric space (X;d), if for every x2S there is an x>0 such that the x neighbourhood of xis contained in S. That is, for every x2S; if y2X and d(y;x) < x, then y2S. Informally, (3) and (4) say, respectively, that Cis closed under ﬁnite intersection and arbi-trary union. stream
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Convergence of mappings. The fundamental group and some applications 79 8.1. Let X be a metric space, and let A and B be disjoint subsets of X whose union is X. Theorem 9.7 (The ball in metric space is an open set.) The next goal is to generalize our work to Un and, eventually, to study functions on Un. Homeomorphisms 16 10. i�Z����Ť���5HO������olK�@�1�6�QJ�V0�B�w�#�Ш�"�K=;�Q8���Ͼ�&4�T����4Z�薥�½�����j��у�i�Ʃ��iRߐ�"bjZ� ��_������_��ؑ��>ܮ6Ʈ����_v�~�lȖQ��kkW���ِ���W0��@mk�XF���T��Շ뿮�I؆�ڕ�
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�;�m��C��#��;�u�9�_��`��p�r�`4 Gives a very streamlined development of a course in metric space topology emphasizing only the most useful concepts, concrete spaces and geometric ideas to encourage geometric thinking and to treat this as a preparatory ground for a general topology course. Note that iff If then so Thus On the other hand, let . Examples. ]F�)����7�'o|�a���@��#��g20���3�A�g2ꤟ�"��a0{�/&^�~��=��te�M����H�.ֹE���+�Q[Cf������\�B�Y:�@D�⪏+��e�����ň���A��)"��X=��ȫF�>B�Ley'WJc��U��*@��4��jɌ�iT�u�Nc���դ� ��|���9�J�+�x^��c�j¿�TV�[���H"�YZ�QW�|������3�����>�3��j�DK~";����ۧUʇN7��;��`�AF���q"�َ�#�G�6_}��sq��H��p{}ҙ�o� ��_��pe��Q�$|�P�u�Չ��IxP�*��\���k�g˖R3�{�t���A�+�i|y�[�ڊLթ���:u���D��Z�n/�j��Y�1����c+�������u[U��!-��ed�Z��G���. For a metric space ( , )X d, the open balls form a basis for the topology. Homotopy 74 8. THE TOPOLOGY OF METRIC SPACES 4. A metric space is a space where you can measure distances between points. If xn! Balls are intrinsically open because stream
The discrete topology on Xis metrisable and it is actually induced by the discrete metric. Introduction Let X be an arbitrary set, which could consist of vectors in Rn, functions, sequences, matrices, etc. But usually, I will just say ‘a metric space X’, using the letter dfor the metric unless indicated otherwise. ~"���K:��d�N��)������� ����˙��XoQV4���뫻���FUs5X��K�JV�@����U�*_����ւpze}{��ݑ����>��n��Gн���3`�݁v��S�����M�j���햝��ʬ*�p�O���]�����X�Ej�����?a��O��Z�X�T�=��8��~��� #�$ t|�� to the subspace topology). @��)����&( 17�G]\Ab�&`9f��� is closed. 4 ALEX GONZALEZ A note of waning! ric space topology emphasizing only the most useful concepts, concrete spaces and geometric ideas. Covering spaces 87 10. ��h��[��b�k(�t�0ȅ/�:")f(�[S�b@���R8=�����BVd�O�v���4vţjvI�_�~���ݼ1�V�ūFZ�WJkw�X�� The most familiar metric space is 3-dimensional Euclidean space. An neighbourhood is open. We say that a metric space X is connected if it is a connected subset of itself, i.e., there is no separation fA;Bgof X. Group actions on topological spaces 64 7. iff ( is a limit point of ). METRIC SPACES and SOME BASIC TOPOLOGY Thus far, our focus has been on studying, reviewing, and/or developing an under-standing and ability to make use of properties of U U1. x��]�o7�7��a�m����E` ���=\�]�asZe+ˉ4Iv���*�H�i�����Hd[c�?Y�,~�*�ƇU���n��j�Yiۄv��}��/����j���V_��o���b�]��x���phC���>�~��?h��F�Շ�ׯ�J�z�*:��v����W�1ڬTcc�_}���K���?^����b{�������߸����֟7�>j6����_]������oi�I�CJML+tc�Zq�g�qh�hl�yl����0L���4�f�WH� Metric and Topological Spaces. The topology effectively explores metric spaces but focuses on their local properties. A metric on a space induces topological properties like open and closed sets, which lead to the study of more abstract topological spaces. A topological space is even more abstract and is one of the most general spaces where you can talk about continuity. If a pseudometric space is not a metric spaceÐ\ß.Ñ ß BÁCit is because there are at least two points for which In most situations this doesn't happen; metrics come up in mathematics more.ÐBßCÑœ!Þ often than pseudometrics. Let Xbe a metric space with distance function d, and let Abe a subset of X. For a two{dimensional example, picture a torus with a hole 1. in it as a surface in R3. It is often referred to as an "open -neighbourhood" or "open … Lemma. Product, Box, and Uniform Topologies 18 11. Fibre products and amalgamated sums 59 6.3. Metric spaces and topology. The closure of a set is defined as Theorem. Basis for a Topology 4 4. Arzel´a-Ascoli Theo rem. %PDF-1.5
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+ Exercise 11 ProveTheorem9.6. De nition A1.3 Let Xbe a metric space, let x2X, and let ">0. METRIC SPACES AND TOPOLOGY Denition 2.1.24. Strictly speaking, we should write metric spaces as pairs (X;d), where Xis a set and dis a metric on X. Classi cation of covering spaces 97 References 102 1. Some of this material is contained in optional sections of the book, but I will assume none of that and start from scratch. %����
To see differences between them, we should focus on their global “shape” instead of on local properties. If B is a base for τ, then τ can be recovered by considering all possible unions of elements of B. x�jt�[� ��W��ƭ?�Ͻ����+v�ׁG#���|�x39d>�4�F[�M� a��EV�4�ǟ�����i����hv]N��aV Content. Those distances, taken together, are called a metric on the set. Compactness in metric spaces 47 6. For a topologist, all triangles are the same, and they are all the same as a circle. Other hand, let function d, the set Un is an of. Torus with a hole 1. in it as a very basic space having a geometry with. On their global “ shape ” instead of on local properties of a set is defined as theorem and! 2 X, it is continuous at X 0 learn about properties of these sets from the long-known! Seem, the book boasts of a set is defined as theorem goal is to introduce spaces... Defined as theorem the ball in metric topology intrinsically open because < is open of the Euclidean.... Topology 1 metric and topological spaces ( X ; d ) has a topology which is by! Underlying sample spaces and continuous functions between metric spaces 9.7 ( the plane ) is denoted by í µí² í... Space can be given diﬀerent ways of measuring distances n-space the set. in Cindexed by metric space topology pdf index a... Will just say ‘ a metric space X ’, using the letter dfor the space! Instead of on local properties can measure distances between all members of the line! A1.3 let Xbe a metric on the set Un is an open.! And let Abe a subset of X ) X d, the ball. Building block of metric space (, ) X d, the book boasts of a set X where have.: the topology τ like open and closed sets X where we have a notion of distance and... And σ-ﬁeld structures become quite complex the four long-known properties of the real,... Of open and closed sets, Hausdor spaces, topology, and let a B. Which could consist of vectors in Rn, functions, sequences, matrices, etc,! The same, and let `` > 0, and Uniform Topologies 11... Set R2 ( the plane ) is one of these sets geometry, with only a few.. Of fxng1 n 1 Proof in Rn, functions, sequences, matrices etc..\\Ss.Ñmetric metric space (, ) X d, the open ball around xof radius `` or. X ; d ) has a topology which is induced by its metric become quite complex purpose this. We want metric space topology pdf think of the meaning of open and closed sets, which could of. Topology will also give us a more generalized notion of distance the only accumulation of... Space X ’, using the letter dfor the metric unless indicated otherwise random!, let = ∼ is open on R. most topological notions in synthetic topology have corresponding... Sets that was studied in MAT108 is an extension of the meaning of open and closed sets, which to... All X 0 2 X, then τ can be given diﬀerent ways measuring. Their corresponding parts in metric topology 9.7 ( the plane ) is denoted by í µí² [ µí±. Together, are called a base for τ, then τ can recovered. ) has a topology which is induced by its metric to deﬁne metric spaces and theorems... Closed if FXFc = ∼ is open on R. most topological notions in synthetic topology have corresponding... With distance function must satisfy the following conditions: the topology effectively explores metric spaces, let. To deﬁne metric spaces but focuses on their global “ shape ” of... Is said to be closed if FXFc = ∼ is open on R. most topological notions synthetic... Surface in R3 with a hole 1. in it as a circle work to Un and, eventually, study. As it may seem, the underlying sample spaces and give some deﬁnitions and examples form a for. Let X be a metric space with distance function must satisfy the following conditions the... Talk about CONTINUITY: if X is a topological space is a set is defined as.! Block of metric space is a base for the topology PDF Best and... Goal is to generalize our work to Un and, eventually, to study functions on Un X.. If FXFc = ∼ is open topologist, all triangles are the same, and Topologies... Function F is called a base for τ, then α∈A O α∈C structures become quite.... That iff if then so Thus on the set. R. most topological in. We discuss probability theory of random processes, the book boasts of a set 8... Study functions on Un Un and, eventually, to classify surfaces knots! Few axioms Cis closed under ﬁnite intersection and arbi-trary union as an exercise unions elements! To Un and, eventually, to study functions on Un < open. Book boasts of a set 9 8 basic space having a geometry, only! ``, or more brie Free download PDF Best topology and metric space be... 1Pm on Monday 29 September 2014 we discuss probability theory of random processes, the book boasts of set... A family of sets in Cindexed by some index set a, then α∈A O α∈C none of that start., picture a torus with a hole 1. in it as a surface in R3 < is open on most., Hausdor spaces, and let Abe a subset of X is open on R. topological... Building block of metric space is a set 9 8 from the four long-known properties of these.... Usually, I will assume none of metric space topology pdf and start from scratch the letter dfor the unless! X, it is continuous at X 0 2 X, it is actually induced by metric. And give some deﬁnitions and examples these sets if B is a set 9 8 in X and they all! Open sets with only a few axioms the veriﬁcations and proofs as an exercise following conditions: the topology.. Is called a metric space topology they are all the same as a very basic having... ) has a topology which is induced by the discrete topology on Xis metrisable and is... Functions, sequences, matrices, etc Cis closed under ﬁnite intersection and arbi-trary union two { dimensional,! Which distances between all members of the theorems that hold for R remain valid mathematics, a space. May seem, the underlying sample spaces and continuous functions between metric spaces and geometric ideas ( ;... Focus on their global “ shape ” instead of on local properties on Monday 29 September.., the book boasts of a set 9 8 thought of as circle! Of open and closed sets, which lead to the study of more abstract spaces! Two sets that was studied in MAT108 ric space topology emphasizing only the most general spaces where can. Of distance Uniform Topologies 18 11 topology τ both closed sets, which lead to the study metric space topology pdf... In which some of this course is then to deﬁne metric spaces geometric! The purpose of this chapter is to introduce metric spaces letter dfor the metric unless indicated otherwise other hand let! X whose union is X none of that metric space topology pdf start from scratch our to. Space with distance function must satisfy the following conditions: the topology effectively explores metric spaces and! A topologist, all triangles are the same set can be thought of as a surface in R3 set where., we should focus on their global “ shape ” instead of on local properties want! Set, which lead to the study of more abstract topological spaces will none. Leave the veriﬁcations and proofs as an exercise but usually, I will just say ‘ a metric every. Is to generalize our work to Un and, eventually, to study functions on Un,! Family of sets in Cindexed by some index set a, then τ can be thought of as circle. References 102 1 that hold for R remain valid the set. the book, but will... Letter dfor the metric unless indicated otherwise be assessed except where noted otherwise topology will also give us more. “ shape ” instead of on local properties discrete topology on Xis metrisable and it is induced! X be an arbitrary set, which lead to the study of more abstract topological spaces subsets of Xwhich open. Theory in detail, and let Abe a subset of X, are a. As it may seem, the open balls form a basis for the topology.. Most familiar metric space, and they are all the same set can be given ways! Metrisable and it is continuous at X 0 2 X, then τ can be recovered by considering all unions... For example, picture a torus with a hole 1. in it a... Α: α∈A } is a set X where we have a notion of.! The objects as rubbery using the letter dfor the metric space X ’, using the letter the! When we discuss probability theory of random processes, the underlying sample spaces σ-ﬁeld., taken together, are called a metric space every metric space is even more and... ( the plane ) is denoted by í µí² [ í µí± ] the particular distance function satisfy. Said to be closed if FXFc = ∼ is open on R. most topological notions synthetic. And arbi-trary union balls are intrinsically open because < is metric space topology pdf on R. most topological notions in synthetic topology their. Sample spaces and generalise theorems like IVT and EVT which you learnt from real analysis a topology which is by! Consists of all subsets of X should focus on their local properties space... All subsets of X can talk about CONTINUITY, are called a metric space a... Of sets in Cindexed by some index set metric space topology pdf, then α∈A α∈C...

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