This distance function 1 Metric spaces IB Metric and Topological Spaces 1 Metric spaces 1.1 De nitions As mentioned in the introduction, given a set X, it is often helpful to have a notion of distance between points. In other words, no sequence may converge to two diﬀerent limits. called a discrete metric; (X;d) is called a discrete metric space. (M2) d( x, y ) = 0 if and only if x = y. Metric Spaces, Open Balls, and Limit Points DEFINITION: A set , whose elements we shall call points, is said to be a metric space if with any two points and of there is associated a real number ( , ) called the distance from to . De nitions, and open sets. Source: daiict.ac.in, Metric Spaces Handwritten Notes A closed subspace of a compact metric space is compact. We will write (X,ρ) to denote the metric space X endowed with a metric ρ. Topology of Metric Spaces: Open and closed ball, Neighborhood, Open set, Interior of a set, Limit point of a set, Derived set, Closed set, Closure of a set, Diameter of a set, Cantor’s theorem, Subspaces, Dense set. Metric Spaces (Notes) These are updated version of previous notes. %����
Metric Spaces The following de nition introduces the most central concept in the course. In these “Metric Spaces Notes PDF”, we will study the concepts of analysis which evidently rely on the notion of distance. endobj
(M3) d( x, y ) = d( y, x ). Introduction Let X … A ball B of radius r around a point x ∈ X is B = {y ∈ X|d(x,y) < r}. In these “Metric Spaces Notes PDF”, we will study the concepts of analysis which evidently rely on the notion of distance. Suppose that Mis a compact metric space and that SˆMis a closed subspace. NOTES FOR MATH 4510, FALL 2010 DOMINGO TOLEDO 1. Deﬁnition 1. Then there is an automatic metric d Y on Y deﬁned by restricting dto the subspace Y× Y, d Y = dY| × Y. Example 7.4. x��]ms�F����7����˻�o�is��䮗i�A��3~I%�m���%e�$d��N]��,�X,��ŗ?O�~�����BϏ��/�z�����.t�����^�e0E4�Ԯp66�*�����/��l��������W�{��{��W�|{T�F�����A�hMi�Q_�X�P����_W�{�_�]]V�x��ņ��XV�t§__�����~�|;_-������O>Φnr:���r�k��_�{'�?��=~��œbj'��A̯ The limit of a sequence in a metric space is unique. 1.2 Open Sets (in a metric space) Now that we have a notion of distance, we can deﬁne what it means to be an open set in a metric space. is complete if it’s complete as a metric space, i.e., if all Cauchy sequences converge to elements of the n.v.s. The third property is called the triangle inequality. These are not the same thing. Let X be a metric space. These are the notes prepared for the course MTH 304 to be o ered to undergraduate students at IIT Kanpur. §1. Topology Generated by a Basis 4 4.1. Topological Spaces 3 3. Deﬁne d: R2 ×R2 → R by d(x,y) = (x1 −y1)2 +(x2 −y2)2 x = (x1,x2), y = (y1,y2).Then d is a metric on R2, called the Euclidean, or ℓ2, metric.It corresponds to Recall that every normed vector space is a metric space, with the metric d(x;x0) = kx x0k. A useful metric on this space is the tree metric, d(x,y) = 1 min{n: xn ̸= yn}. TOPOLOGY: NOTES AND PROBLEMS Abstract. <>/Font<>/XObject<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 612 792] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>>
Metric Spaces A metric space is a set X endowed with a metric ρ : X × X → [0,∞) that satisﬁes the following properties for all x, y, and z in X: 1. ρ(x,y) = 0 if and only if x = y, 2. ρ(x,y) = ρ(y,x), and 3. ρ(x,z) ≤ ρ(x,y)+ ρ(y,z). 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. ���A��..�O�b]U*�
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