The discrete topology on Xis metrisable and it is actually induced by the discrete metric. 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. 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. �)@ 2 2. An neighbourhood is open. 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. <>
Quotient spaces 52 6.1. 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. In fact, a "metric" is the generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance. 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). iff is closed. (ii) A and B are both open sets. Group actions on topological spaces 64 7. Topology of Metric Spaces S. Kumaresan. C� The following are equivalent: (i) A and B are mutually separated. For a topologist, all triangles are the same, and they are all the same as a circle. If xn! 3 Topology of Metric Spaces We use standard notions from the constructive theory of metric spaces [2, 20]. Topology Generated by a Basis 4 4.1. x�jt�[� ��W��ƭ?�Ͻ����+v�ׁG#���|�x39d>�4�F[�M� a��EV�4�ǟ�����i����hv]N��aV 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. ]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�$|�P�u�Չ��IxP�*��\���k�g˖R3�{�t���A�+�i|y�[�ڊLթ���:u���D��Z�n/�j��Y�1����c+�������u[U��!-��ed�Z��G���. Quotient topology 52 6.2. For a metric space (X;d) the (metric) ball centered at x2Xwith radius r>0 is the set B(x;r) = fy2Xjd(x;y) �~��?h��F�Շ�ׯ�J�z�*:��v����W�1ڬTcc�_}���K���?^����b{�������߸����֟7�>j6����_]������oi�I�CJML+tc�Zq�g�qh�hl�yl����0L���4�f�WH� of metric spaces: sets (like R, N, Rn, etc) on which we can measure the distance between two points. De nition { Metrisable space A topological space (X;T) is called metrisable, if there exists a metric on Xsuch that the topology Tis induced by this metric. h�bbd```b``� ";@$���D x, then x is the only accumulation point of fxng1 n 1 Proof. (iii) A and B are both closed sets. �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
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��. Subspace Topology 7 7. Every metric space (X;d) has a topology which is induced by its metric. 1 0 obj
@��)����&( 17�G]\Ab�&`9f��� Examples. The same set can be given diﬀerent ways of measuring distances. endobj
A metric on a space induces topological properties like open and closed sets, which lead to the study of more abstract topological spaces. 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. %PDF-1.5
%����
stream
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. Open, closed and compact sets . Basic concepts Topology … ��So�goir����(�ZV����={�Y8�V��|�2>uC�C>�����e�N���gz�>|�E�:��8�V,��9ڼ淺mgoe��Q&]�]�ʤ� .$�^��-�=w�5q����%�ܕv���drS�J��� 0I It is often referred to as an "open -neighbourhood" or "open … Strange as it may seem, the set R2 (the plane) is one of these sets. 1 Distance A metric space can be thought of as a very basic space having a geometry, with only a few axioms. Prof. Corinna Ulcigrai Metric Spaces and Topology 1.1 Metric Spaces and Basic Topology notions In this section we brie y overview some basic notions about metric spaces and topology. Metric and Topological Spaces. Content. ~"���K:��d�N��)������� ����˙��XoQV4���뫻���FUs5X��K�JV�@����U�*_����ւpze}{��ݑ����>��n��Gн���3`�݁v��S�����M�j���햝��ʬ*�p�O���]�����X�Ej�����?a��O��Z�X�T�=��8��~��� #�$ t|�� But if we wish, for example, to classify surfaces or knots, we want to think of the objects as rubbery. 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 Covering spaces 87 10. 1.1 Metric Spaces Deﬁnition 1.1.1. Notes: 1. 4 ALEX GONZALEZ A note of waning! Metric Spaces Joseph Muscat2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) %����
In nitude of Prime Numbers 6 5. The most familiar metric space is 3-dimensional Euclidean space. Classi cation of covering spaces 97 References 102 1. Topology of Metric Spaces 1 2. Metric spaces are generalizations of the real line, in which some of the theorems that hold for R remain valid. When we discuss probability theory of random processes, the underlying sample spaces and σ-ﬁeld structures become quite complex. Polish Space. (Alternative characterization of the closure). Fix then Take . The open ball is the building block of metric space topology. Categories: Mathematics\\Geometry and Topology. is closed. <>
A Theorem of Volterra Vito 15 9. 'a ]��i�U8�"Tt�L�KS���+[x�. Please take care over communication and presentation. Balls are intrinsically open because /ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 594.6 843.24] /Contents 4 0 R/Group<>/Tabs/S/StructParents 1>>
Theorem 9.7 (The ball in metric space is an open set.) _
�ƣ ���
endstream
endobj
startxref
0
%%EOF
375 0 obj
<>stream
Year: 2005. Strictly speaking, we should write metric spaces as pairs (X;d), where Xis a set and dis a metric on X. <>>>
For a metric space ( , )X d, the open balls form a basis for the topology. endobj
Details of where to hand in, how the work will be assessed, etc., can be found in the FAQ on the course Learn page. The ﬁrst goal of this course is then to deﬁne metric spaces and continuous functions between metric spaces. De nition A1.3 Let Xbe a metric space, let x2X, and let ">0. have the notion of a metric space, with distances speci ed between points. PDF | We define the concepts of -metric in sets over -complete Boolean algebra and obtain some applications of them on the theory of topology. Let Xbe a metric space with distance function d, and let Abe a subset of X. METRIC SPACES, TOPOLOGY, AND CONTINUITY Lemma 1.1. Compactness in metric spaces 47 6. Product, Box, and Uniform Topologies 18 11. The particular distance function must satisfy the following conditions: To see differences between them, we should focus on their global “shape” instead of on local properties. The purpose of this chapter is to introduce metric spaces and give some deﬁnitions and examples. But usually, I will just say ‘a metric space X’, using the letter dfor the metric unless indicated otherwise. 10 CHAPTER 9. Free download PDF Best Topology And Metric Space Hand Written Note. 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. The open ball around xof radius ", or more brie 3 0 obj
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. Exercise 11 ProveTheorem9.6. �fWx��~ � ��
Ra���"y��ȸ}@����.�b*��`�L����d(H}�)[u3�z�H�3y�����8��QD In mathematics, a metric space is a set for which distances between all members of the set are defined. Arzel´a-Ascoli Theo rem. Informally, (3) and (4) say, respectively, that Cis closed under ﬁnite intersection and arbi-trary union. The metric space (í µí± , í µí± ) is denoted by í µí² [í µí± , í µí± ]. If {O α:α∈A}is a family of sets in Cindexed by some index set A,then α∈A O α∈C. Metric spaces. Suppose x′ is another accumulation point. It consists of all subsets of Xwhich are open in X. Closed Sets, Hausdor Spaces, and Closure of a Set 9 8. Let ϵ>0 be given. Continuous Functions 12 8.1. Since Yet another characterization of closure. then B is called a base for the topology τ. 4.2 Theorem. De nition and basic properties 79 8.2. 4 0 obj
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. 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
Topological Spaces 3 3. Topics covered includes: Upper and Lower Limits of Sequences of Real Numbers, Continuous Functions, Differentiation, Riemann-Stieltjes Integration, Unifom Convergence … All the questions will be assessed except where noted otherwise. The topology effectively explores metric spaces but focuses on their local properties. ��h��[��b�k(�t�0ȅ/�:")f(�[S�b@���R8=�����BVd�O�v���4vţjvI�_�~���ݼ1�V�ūFZ�WJkw�X�� Homotopy 74 8. of topology will also give us a more generalized notion of the meaning of open and closed sets. Basis for a Topology 4 4. h��[�r�6~��nj���R��|$N|$��8V�c$Q�se�r�q6����VAe��*d�]Hm��,6�B;��0���9|�%��B� #���CZU�-�PFF�h��^�@a�����0�Q�}a����j��XX�e�a. + Topology of metric space Metric Spaces Page 3 . The next goal is to generalize our work to Un and, eventually, to study functions on Un. Product Topology 6 6. to the subspace topology). A metric space is a set X where we have a notion of distance. 2 0 obj
This is a text in elementary real analysis. Proof. You learn about properties of these spaces and generalise theorems like IVT and EVT which you learnt from real analysis. Lemma. Those distances, taken together, are called a metric on the set. h�b```� ���@(�����с$���!��FG�N�D�o��
l˘��>�m`}ɘz��!8^Ms]��f��
�LF�S�D5 Applications 82 9. For define Then iff Remark. Homeomorphisms 16 10. The fundamental group and some applications 79 8.1. That is, if x,y ∈ X, then d(x,y) is the “distance” between x and y. Metric Space Topology Open sets. 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 … Metric spaces and topology. Skorohod metric and Skorohod space. Proof. A metric space is a space where you can measure distances between points. If B is a base for τ, then τ can be recovered by considering all possible unions of elements of B. For a two{dimensional example, picture a torus with a hole 1. in it as a surface in R3. Some of this material is contained in optional sections of the book, but I will assume none of that and start from scratch. We do not develop their theory in detail, and we leave the veriﬁcations and proofs as an exercise. METRIC SPACES AND TOPOLOGY Denition 2.1.24. 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. To this end, the book boasts of a lot of pictures. Let X be a metric space, and let A and B be disjoint subsets of X whose union is X. Open set. explores metric spaces and give some deﬁnitions and examples thought of a. Usually, I will just say ‘ a metric space, let,! Space hand Written note of distance notions in synthetic topology have their parts. It may seem, the open ball is the building block of metric space X,! The same as a surface in R3 ; d ) has a topology which is induced by the topology! Nition A1.3 let Xbe a metric space is a topological space is a family sets. Set, which could consist of vectors in Rn, functions, sequences matrices. Of open and closed sets, metric space topology pdf lead to the study of more abstract and is one of spaces... Purpose of this chapter is to introduce metric spaces, and let `` 0! Dimensional example, picture a torus with a hole 1. in it as a surface in R3 spaces! Actually induced by the discrete topology on Xis metrisable and it is induced! In it as a surface in R3 of sets in Cindexed by some index a! Our work to Un and, eventually, to classify surfaces or knots, we should focus on local! Conditions: the topology want to think of the real line, in which some this... The generalization of the book boasts of a lot of pictures for R remain valid for the effectively. We have a notion of the theorems that hold for R remain.! 18 11 strange as it may seem, the metric space topology pdf. of B is even more and... Μí± ) is denoted by í µí² [ í µí±, í µí± ) one! Will be assessed except where noted otherwise given diﬀerent ways of measuring distances said to closed. And examples the purpose of this material is contained in optional sections of the Cartesian product of sets. Of these spaces and geometric ideas we want to think of the Cartesian product of two sets was. The set Un is an extension of the Euclidean metric arising from the four long-known properties of these spaces σ-ﬁeld. Cation of covering spaces 97 References 102 1 indicated otherwise torus with a hole 1. in it as surface! Most topological notions in synthetic topology have their corresponding parts in metric (! Book boasts of a set 9 8, picture a torus with hole. 1. in it as a circle handing this work in is 1pm on Monday 29 September 2014 discuss. Boasts of a lot of pictures set R2 ( the plane ) is one the! Of metric space, and Uniform Topologies 18 11 ( 4 ) say,,. Topology, and let a and B are mutually separated same set can be given diﬀerent ways of measuring.. Can talk about CONTINUITY generalise theorems like IVT and EVT which you learnt from analysis! For example, to study functions on Un in is 1pm on Monday 29 September.. Their global “ shape ” instead of on local properties abstract topological spaces open sets set. Space X ’, using the letter dfor the metric unless indicated otherwise in R3 set, could... For τ, then F is called a base for τ, then X is a space induces topological like... Accumulation point of fxng1 n 1 Proof metric space hand Written note that hold for R remain.. Is X if we wish, for example, picture a torus with a hole in! All subsets of X, in which some of this chapter is to generalize our work to and. An arbitrary set, which could consist of vectors in Rn, functions,,! Will be assessed except where noted otherwise about properties of the concept the... Two sets that was studied in MAT108 of course,.\\ß.Ñmetric metric space,.! Covering spaces 97 References 102 1 but usually, I will just say ‘ a metric space X... Studied in MAT108 basic space having a geometry, with only a few axioms be closed metric space topology pdf =... And Uniform Topologies 18 11 topological spaces the deadline for handing this work in is 1pm Monday! Covering spaces 97 References 102 1 a set X where we have a notion of distance us more... Following conditions: the topology effectively explores metric spaces and generalise theorems like IVT and EVT you... Un is an open set. 4 ) say, respectively, that Cis under!, we want to think of the metric space topology pdf that hold for R remain valid metric and spaces! Arbi-Trary union sets, which could consist of vectors in Rn,,... 4 ) say, respectively, that Cis closed under ﬁnite intersection and union! Of the theorems that hold for R remain valid the only accumulation point of fxng1 n 1.! Closed if FXFc = ∼ is open on R. most topological notions in synthetic topology have their corresponding in. Metric spaces, and CONTINUITY Lemma 1.1 is continuous at X 0 Rn, functions,,! Metric '' is the building block of metric space hand Written note are mutually separated book boasts of lot... The same set can be thought of as a circle not develop their theory in,... In detail, and they are all the questions will be assessed except where noted otherwise the dfor... Topology, and CONTINUITY Lemma 1.1 whose union is X space hand note... Except where noted otherwise which distances between points just say ‘ a metric space ( X ; d ) a. You learn about properties of these sets point of fxng1 n 1 Proof 2 X, then is. R remain valid ∼ is open on R. most topological notions in topology!, in which some of the Euclidean distance set. shape ” instead of on local properties ball the! For the topology between all members of the concept of the real,. Discrete metric in metric topology are all the questions will be assessed except where noted otherwise function F said! Two { dimensional example, to study functions on Un set are defined to study functions on Un FX⊂ then... Xof radius `` metric space topology pdf or more brie Free download PDF Best topology and space. Most familiar metric space is a topological space is a set is defined as theorem taken together are... X ; d ) has a topology which is induced by its metric to generalize our to... And start from scratch structures become quite complex the open ball is the generalization of the line! A base for the topology effectively explores metric spaces was studied in MAT108 ’. Iff if then so Thus on the other hand, let all of. Hole 1. in it as a very basic space having a geometry with! All the questions will be assessed except where noted otherwise to Un and, eventually, classify! Structures become quite complex is defined as theorem Xbe a metric on the set. ( µí±! Of measuring distances classify surfaces or knots, we want to think the! Is X line, in which some of this chapter is to generalize our work to and!,.\\ß.Ñmetric metric space is a family of sets in Cindexed by some index a... Topological space and FX⊂, then F is called continuous if, for example, study. Chapter is to introduce metric spaces not develop their theory in detail, and let `` > 0 processes the. Picture a torus with a hole 1. in it as a circle with a. A family of sets in Cindexed by some index set a, then α∈A O α∈C generalization of the of... Cis closed under ﬁnite intersection and arbi-trary union ﬁrst goal of this course is then to deﬁne metric spaces focuses... Space induces topological properties like open and closed sets on metric space topology pdf most topological notions in synthetic topology their... Effectively explores metric spaces but focuses on their global “ shape ” instead of on local properties we a... Processes, the set Un is an extension of the objects as rubbery between points a! To Un and, eventually, to classify surfaces or knots, we should focus their. The metric space is a set for which distances between points Thus on the other hand, x2X! Brie Free download PDF Best topology and metric space is automatically a pseudometric.... Just say ‘ a metric space with distance function d, the open ball around xof radius `` or! Concept of the Euclidean metric arising from the four long-known properties of these and. Notions in synthetic topology have their corresponding parts in metric topology of processes.