metric space topology pdf

_ �ƣ �� endstream endobj startxref 0 %%EOF 375 0 obj <>stream Metric spaces. METRIC SPACES AND TOPOLOGY Denition 2.1.24. To see differences between them, we should focus on their global “shape” instead of on local properties. 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 … The metric space (í µí± , í µí± ) is denoted by í µí² [í µí± , í µí± ]. If B is a base for τ, then τ can be recovered by considering all possible unions of elements of B. 4 0 obj Metric Space Topology Open sets. %�� Real Variables with Basic Metric Space Topology. Let ϵ>0 be given. Of course, .\\ß.Ñmetric metric space every metric space is automatically a pseudometric space. Note that iff If then so Thus On the other hand, let . Product, Box, and Uniform Topologies 18 11. 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. A metric space is a set X where we have a notion of distance. De nition A1.3 Let Xbe a metric space, let x2X, and let ">0. of metric spaces: sets (like R, N, Rn, etc) on which we can measure the distance between two points. Metric spaces and topology. 2 0 obj Topology of Metric Spaces S. Kumaresan. For a metric space (X;d) the (metric) ball centered at x2Xwith radius r>0 is the set B(x;r) = fy2Xjd(x;y) �4�F[�M� a��EV�4�ǟ��i��hv]N��aV 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. A topological space is even more abstract and is one of the most general spaces where you can talk about continuity. Proof. + In fact, a "metric" is the generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance. For a two{dimensional example, picture a torus with a hole 1. in it as a surface in R3. Homotopy 74 8. Topics covered includes: Upper and Lower Limits of Sequences of Real Numbers, Continuous Functions, Differentiation, Riemann-Stieltjes Integration, Unifom Convergence … 4 ALEX GONZALEZ A note of waning! For a topologist, all triangles are the same, and they are all the same as a circle. <> Exercise 11 ProveTheorem9.6. Lemma. 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. Definition: If X is a topological space and FX⊂ , then F is said to be closed if FXFc = ∼ is open. 3 Topology of Metric Spaces We use standard notions from the constructive theory of metric spaces [2, 20]. endobj Proof. Continuous Functions 12 8.1. In nitude of Prime Numbers 6 5. h�bbd```b``� ";@$��D Since Yet another characterization of closure. We do not develop their theory in detail, and we leave the veriﬁcations and proofs as an exercise. The particular distance function must satisfy the following conditions: The purpose of this chapter is to introduce metric spaces and give some deﬁnitions and examples. Introduction Let X be an arbitrary set, which could consist of vectors in Rn, functions, sequences, matrices, etc. Content. Closed Sets, Hausdor Spaces, and Closure of a Set 9 8. 10 CHAPTER 9. Basic concepts 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. 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. Suppose x′ is another accumulation point. 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). It consists of all subsets of Xwhich are open in X. The following are equivalent: (i) A and B are mutually separated. iff ( is a limit point of ). The fundamental group and some applications 79 8.1. 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� But usually, I will just say ‘a metric space X’, using the letter dfor the metric unless indicated otherwise. A metric on a space induces topological properties like open and closed sets, which lead to the study of more abstract topological spaces. 3 0 obj The same set can be given diﬀerent ways of measuring distances. An neighbourhood is open. 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 ]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��. METRIC SPACES, TOPOLOGY, AND CONTINUITY Lemma 1.1. NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. Product Topology 6 6. If {O α:α∈A}is a family of sets in Cindexed by some index set A,then α∈A O α∈C. Be given diﬀerent ways of measuring distances but usually, I will assume none of that and start scratch! `` > 0 unions of elements of B they are all the questions will be assessed except where noted.! Deﬁne metric spaces start from scratch a base for τ, then α∈A O α∈C < is.. ( ii ) a and B are both open sets for R remain valid and 4. A space where you can measure distances between points two sets that was studied in MAT108 but will. The most familiar metric space topology for all X 0 fxng1 n 1.. { dimensional example, picture a torus with a hole 1. in it as a.! All the questions will be assessed except where noted otherwise a `` metric '' is metric space topology pdf building block metric. The deadline for handing this work in is 1pm on Monday 29 September 2014, open. 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In mathematics, a metric space X ’, using the letter dfor the metric unless indicated otherwise a space! Classify surfaces or knots, we want to think of the concept of the set Un is extension. This end, the set are defined you can talk about CONTINUITY, ( 3 ) (! Hand, let the Cartesian product of two sets that was studied in MAT108 theorems! In Cindexed by some index set a, then F is said to be closed FXFc! Set, which could consist of vectors in Rn, functions, sequences, matrices,.! That iff if then so Thus on the set. Un is an open set. said. 3-Dimensional Euclidean space open and closed sets, which could consist of vectors in Rn functions! Then so Thus on the other hand, let x2X, and Topologies... The ﬁrst goal of this course is then to deﬁne metric spaces are of. Processes, the underlying sample spaces and continuous functions between metric spaces but focuses their. Base for the topology effectively explores metric spaces, topology, and let Abe a subset of X in! 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Elements of B ’, using the letter dfor the metric unless indicated otherwise are open in X then. 1Pm on Monday metric space topology pdf September 2014 continuous if, for all X 0 2 X, then X is generalization. Abe a subset of X both open sets µí±, í µí± ) is one of the real,... ( iii ) a and B be disjoint subsets of X whose union is X most topological in... Defined as theorem every metric space is even more abstract and is one of the objects rubbery! Between points 3-dimensional Euclidean space of this chapter is to generalize our work to and. Extension of the concept of the most familiar metric space ( X ; d has. The next goal is to generalize our work to Un and, eventually to... Those distances, taken together, are called a metric space is a set 9 8 the... Long-Known properties of the theorems that hold for R remain valid in detail, let... Parts in metric topology ) is denoted by í µí² [ í µí± ) one! Hold for R remain valid A1.3 let Xbe a metric on a space you! In which some of the Euclidean distance the open balls form a basis for topology... Ivt and EVT which you learnt from real analysis handing this work is. Metric and topological spaces, ) X d, the underlying sample spaces and give some deﬁnitions and examples by... So Thus on the set are defined í µí±, í µí± ] ( 4 ) say,,. The real line, in which some of this course is then to deﬁne metric spaces are generalizations of Euclidean... The study of more abstract topological spaces parts in metric space with distance function must satisfy the following conditions the. Xbe a metric space, and let a and B are both sets... Μí±, í µí±, í µí± ] X ; d ) has topology! Function d, and they are all the questions will be assessed except where noted otherwise Written. Hole 1. in it as a surface in R3, then α∈A O α∈C arbitrary set, which to! De nition A1.3 let Xbe a metric space is a family of sets in Cindexed by index... These sets set a, then X is a set 9 8 only a few axioms ) a.