A metric space is a set X where we have a notion of distance. Closed Sets, Hausdor Spaces, and Closure of a Set 9 8. 1 Distance A metric space can be thought of as a very basic space having a geometry, with only a few axioms. That is, if x,y ∈ X, then d(x,y) is the “distance” between x and y. 3 Topology of Metric Spaces We use standard notions from the constructive theory of metric spaces [2, 20]. 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. The fundamental group and some applications 79 8.1. Topology of Metric Spaces 1 2. Note that iff If then so Thus On the other hand, let . Introduction Let X be an arbitrary set, which could consist of vectors in Rn, functions, sequences, matrices, etc. Homotopy 74 8. The next goal is to generalize our work to Un and, eventually, to study functions on Un. The purpose of this chapter is to introduce metric spaces and give some definitions and examples. The particular distance function must satisfy the following conditions: For a metric space ( , )X d, the open balls form a basis for the topology. 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. Real Variables with Basic Metric Space Topology. For a two{dimensional example, picture a torus with a hole 1. in it as a surface in R3. h��[�r�6~��nj���R��|$N|$��8V�c$Q�se�r�q6����VAe��*d�]Hm��,6�B;��0���9|�%��B� #���CZU�-�PFF�h��^�@a�����0�Q�}a����j��XX�e�a. PDF | We define the concepts of -metric in sets over -complete Boolean algebra and obtain some applications of them on the theory of topology. METRIC SPACES, TOPOLOGY, AND CONTINUITY Lemma 1.1. Since Yet another characterization of closure. Those distances, taken together, are called a metric on the set. Let ϵ>0 be given. 2 0 obj Of course, .\\ß.Ñmetric metric space every metric space is automatically a pseudometric space. 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. It is often referred to as an "open -neighbourhood" or "open … Metric Spaces Joseph Muscat2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) De nition A1.3 Let Xbe a metric space, let x2X, and let ">0. Metric Spaces A metric space is a set X that has a notion of the distance d(x,y) between every pair of points x,y ∈ X. Content. Applications 82 9. Metric spaces are generalizations of the real line, in which some of the theorems that hold for R remain valid. Continuous Functions 12 8.1. All the questions will be assessed except where noted otherwise. A metric on a space induces topological properties like open and closed sets, which lead to the study of more abstract topological spaces. Every metric space (X;d) has a topology which is induced by its metric. Metric spaces and topology. It consists of all subsets of Xwhich are open in X. A topological space is even more abstract and is one of the most general spaces where you can talk about continuity. �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 Details of where to hand in, how the work will be assessed, etc., can be found in the FAQ on the course Learn page. You learn about properties of these spaces and generalise theorems like IVT and EVT which you learnt from real analysis. A Theorem of Volterra Vito 15 9. 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. (Alternative characterization of the closure). Covering spaces 87 10. 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. 4 0 obj ric space topology emphasizing only the most useful concepts, concrete spaces and geometric ideas. Topology of Metric Spaces S. Kumaresan. _ �ƣ ��� endstream endobj startxref 0 %%EOF 375 0 obj <>stream Please take care over communication and presentation. 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. THE TOPOLOGY OF METRIC SPACES 4. %PDF-1.5 �?��Ԃ{�8B���x��W�MZ?f���F��7��_�ޮ�w��7o�y��И�j�qj�Lha8�j�/� /\;7 �3p,v Suppose x′ is another accumulation point. If B is a base for τ, then τ can be recovered by considering all possible unions of elements of B. of topology will also give us a more generalized notion of the meaning of open and closed sets. Strange as it may seem, the set R2 (the plane) is one of these sets. Basis for a Topology 4 4. ��h��[��b�k(�t�0ȅ/�:")f(�[S�b@���R8=�����BVd�O�v���4vţjvI�_�~���ݼ1�V�ūFZ�WJkw�X�� stream <> Balls are intrinsically open because 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���. If {O α:α∈A}is a family of sets in Cindexed by some index set A,then α∈A O α∈C. We do not develop their theory in detail, and we leave the verifications and proofs as an exercise. 1 0 obj The function f is called continuous if, for all x 0 2 X , it is continuous at x 0. 4.2 Theorem. ��So�goir����(�ZV����={�Y8�V��|�2>uC�C>�����e�N���gz�>|�E�:��8�V,��9ڼ淺mgoe��Q&]�]�ʤ� .$�^��-�=w�5q����%�ܕv���drS�J��� 0I Metric and Topological Spaces. An neighbourhood is open. If xn! 'a ]��i�U8�"Tt�L�KS���+[x�. Compactness in metric spaces 47 6. In fact, a "metric" is the generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance. Informally, (3) and (4) say, respectively, that Cis closed under finite intersection and arbi-trary union. ~"���K:��d�N��)������� ����˙��XoQV4���뫻���FUs5X��K�JV�@����U�*_����ւpze}{��ݑ����>��n��Gн���3`�݁v��S�����M�j���햝��ʬ*�p�O���]�����X�Ej�����?a��O��Z�X�T�=��8��~��� #�$ t|�� Metric spaces and topology. The first goal of this course is then to define metric spaces and continuous functions between metric spaces. Open, closed and compact sets . 3 0 obj <> have the notion of a metric space, with distances speci ed between points. Homeomorphisms 16 10. x, then x is the only accumulation point of fxng1 n 1 Proof. endobj For a topologist, all triangles are the same, and they are all the same as a circle. Topological Spaces 3 3. If is closed, then . Proof. endobj <>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 594.6 843.24] /Contents 4 0 R/Group<>/Tabs/S/StructParents 1>> �)@ h�bbd```b``� ";@$���D Group actions on topological spaces 64 7. The most familiar metric space is 3-dimensional Euclidean space. A metric space is a space where you can measure distances between points. 256 0 obj <> endobj 280 0 obj <>/Filter/FlateDecode/ID[<0FF804C6C1889832F42F7EF20368C991><61C4B0AD76034F0C827ADBF79E6AB882>]/Index[256 120]/Info 255 0 R/Length 124/Prev 351177/Root 257 0 R/Size 376/Type/XRef/W[1 3 1]>>stream � �� Ra���"y��ȸ}@����.�b*��`�L����d(H}�)[u3�z�H�3y�����8��QD We want to endow this set with a metric; i.e a way to measure distances between elements of X.A distanceor metric is a function d: X×X →R such that if we take two elements x,y∈Xthe number d(x,y) gives us the distance between them. The discrete topology on Xis metrisable and it is actually induced by the discrete metric. Notes: 1. Basic concepts Topology … 10 CHAPTER 9. Year: 2005. To this end, the book boasts of a lot of pictures. %PDF-1.5 %���� The closure of a set is defined as Theorem. is closed. Therefore, it becomes completely ineffective when the space is discrete (consists of isolated points) However, these discrete metric spaces are not always identical (e.g., Z and Z 2). Exercise 11 ProveTheorem9.6. Product, Box, and Uniform Topologies 18 11. Lemma. Mn�qn�:�֤���u6� 86��E1��N�@����{0�����S��;nm����==7�2�N�Or�ԱL�o�����UGc%;�p�{�qgx�i2ը|����ygI�I[K��A�%�ň��9K# ��D���6�:!�F�ڪ�*��gD3���R���QnQH��txlc�4�꽥�ƒ�� ��W p��i�x�A�r�ѵTZ��X��i��Y����D�a��9�9�A�p�����3��0>�A.;o;�X��7U9�x��. (ii) A and B are both open sets. Product Topology 6 6. i�Z����Ť���5HO������olK�@�1�6�QJ�V0�B�w�#�Ш�"�K=;�Q8���Ͼ�&4�T����4Z�薥�½�����j��у�i�Ʃ��iRߐ�"bjZ� ��_������_��ؑ��>ܮ6Ʈ����_v�~�lȖQ��kkW���ِ���W0��@mk�XF���T��Շ뿮�I؆�ڕ� Cj��- �u��j;���mR�3�R�e!�V��bs1�'�67�Sڄ�;��JiY���ִ��E��!�l��Ԝ�4�P[՚��"�ش�U=�t��5�U�_:|��Q�9"�����9�#���" ��H�ڙ�×[��q9����ȫJ%_�k�˓�������)��{���瘏�h ���킋����.��H0��"�8�Cɜt�"�Ki����.R��r ������a�$"�#�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 Quotient topology 52 6.2. 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� Convergence of mappings. Skorohod metric and Skorohod space. Topology of metric space Metric Spaces Page 3 . iff is closed. 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. Metric spaces. Classi cation of covering spaces 97 References 102 1. endobj General Topology 1 Metric and topological spaces The deadline for handing this work in is 1pm on Monday 29 September 2014. Arzel´a-Ascoli Theo­ rem. 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. NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. Free download PDF Best Topology And Metric Space Hand Written Note. Proof. x�jt�[� ��W��ƭ?�Ͻ����+v�ׁG#���|�x39d>�4�F[�M� a��EV�4�ǟ�����i����hv]N��aV (iii) A and B are both closed sets. 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. Quotient spaces 52 6.1. Let X be a metric space, and let A and B be disjoint subsets of X whose union is X. C� 2 2. Categories: Mathematics\\Geometry and Topology. 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. Fix then Take . But usually, I will just say ‘a metric space X’, using the letter dfor the metric unless indicated otherwise. 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. %���� 4 ALEX GONZALEZ A note of waning! For a metric space (X;d) the (metric) ball centered at x2Xwith radius r>0 is the set B(x;r) = fy2Xjd(x;y) 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. �fWx��~ then B is called a base for the topology τ. 1.1 Metric Spaces Definition 1.1.1. Polish Space. @��)����&( 17�G]\Ab�&`9f��� De nition and basic properties 79 8.2. The following are equivalent: (i) A and B are mutually separated. To see differences between them, we should focus on their global “shape” instead of on local properties. Fibre products and amalgamated sums 59 6.3. h�b```� ���@(�����с$���!��FG�N�D�o�� l˘��>�m`}ɘz��!8^Ms]��f�� �LF�S�D5 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. The open ball is the building block of metric space topology. <>>> Theorem 9.7 (The ball in metric space is an open set.) METRIC SPACES AND TOPOLOGY Denition 2.1.24. In mathematics, a metric space is a set for which distances between all members of the set are defined. In nitude of Prime Numbers 6 5. Is to generalize our work to Un and, eventually, to study functions on Un studied! 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