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�� l˘��>�m`}ɘz��!8^Ms]��f�� �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& }��] endstream endobj 257 0 obj <> endobj 258 0 obj <> endobj 259 0 obj <> endobj 260 0 obj <>stream It consists of all subsets of Xwhich are open in X. <>>> 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 All the questions will be assessed except where noted otherwise. <> 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 have the notion of a metric space, with distances speci ed between points. 3 0 obj C� %PDF-1.5 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� 86��E1��N�@����{0�����S��;nm����==7�2�N�Or�ԱL�o�����UGc%;�p�{�qgx�i2ը|����ygI�I[K��A�%�ň��9K# ��D���6�:!�F�ڪ�*��gD3���R���QnQH��txlc�4�꽥�ƒ�� ��W 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 We do not develop their theory in detail, and we leave the veriﬁcations and proofs as an exercise. 256 0 obj <> endobj 280 0 obj <>/Filter/FlateDecode/ID[<0FF804C6C1889832F42F7EF20368C991><61C4B0AD76034F0C827ADBF79E6AB882>]/Index[256 120]/Info 255 0 R/Length 124/Prev 351177/Root 257 0 R/Size 376/Type/XRef/W[1 3 1]>>stream 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� Cj��- �u��j;���mR�3�R�e!�V��bs1�'�67�Sڄ�;��JiY���ִ��E��!�l��Ԝ�4�P[՚��"�ش�U=�t��5�U�_:|��Q�9"�����9�#���" ��H�ڙ�×[��q9����ȫJ%_�k�˓�������)��{���瘏�h ���킋����.��H0��"�8�Cɜt�"�Ki����.R��r ������a�\$"�#�B�\$KcE]Is��C��d)bN�4����x2t�>�jAJ���x24^��W�9L�,)^5iY��s�KJ���,%�"�5���2�>�.7fQ� 3!�t�*�"D��j�z�H����K�Q�ƫ'8G���\N:|d*Zn~�a�>F��t���eH�y�b@�D���� �ߜ Q�������F/�]X!�;��o�X�L���%����%0��+��f����k4ؾ�۞v��,|ŷZ���[�1�_���I�Â�y;\�Qѓ��Џ�`��%��Kz�Y>���5��p�m����ٶ ��vCa�� �;�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 %���� �?��Ԃ{�8B���x��W�MZ?f���F��7��_�ޮ�w��7o�y��И�j�qj�Lha8�j�/� /\;7 �3p,v <> + 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. 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