topology of metric spaces pdf

More Note that iff If then so Thus On the other hand, let . Let M be an arbitrary metric space. This particular topology is said to be induced by the metric. Remark 1.1.11. De nition (Metric space). y. Let M be an arbitrary metric space. Metric Spaces, Topological Spaces, and Compactness Proposition A.6. A metric space is, essentially, a set of points together with a rule for saying how far apart two such points are: De nition 1.1. Prove that there is a topology on Xconsisting of the co nite subsets together with ;. 2. The other is to pass to the metric space (T;d T) and from there to a metric topology on T. The idea is that both give the same topology T T.] Definition 1.1.10. Then the closed ball of center p, radius r; that is, the set {q ∈ M: d(q,p) ≤ r} is closed. De¿nition 3.2.2 A metric space consists of a pair S˛d –a set, S, and a metric, d, on S. Remark 3.2.3 There are three commonly used (studied) metrics for the set UN. This distance function is known as the metric. De nitions, and open sets. In 1952, a new str uctur e of metric spaces, so called B-metric space was introduced b y Ellis and Sprinkle 2 ,o nt h es e t X to B oolean algebra for deta ils see 3 , 4 . 78 CHAPTER 3. 1. iff ( is a limit point of ). Proof. The set of real numbers R with the function d(x;y) = jx yjis a metric space. Picture for a closed set: Fcontains all of its ‘boundary’ points. Let Xbe a compact metric space. Topology of metric space Metric Spaces Page 3 . 1 THE TOPOLOGY OF METRIC SPACES 3 1. A subset U of X is co nite if XnU is nite. 5.Let X be a set. (Alternative characterization of the closure). If (A) holds, (xn) has a convergent subsequence, xn k! A metric space is a set Xtogether with a metric don it, and we will use the notation (X;d) for a metric space. Pick xn 2 Kn. a metric space, it is only the small distances that matter. The closure of a set is defined as Theorem. Assume K1 ˙ K2 ˙ K3 ˙ form a decreasing sequence of closed subsets of X. A metric space is a non-empty set equi pped with structure determined by a well-defin ed notion of distan ce. Fix then Take . If each Kn 6= ;, then T n Kn 6= ;. Let (X;T ) be a topological space. 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. Let p ∈ M and r ≥ 0. The elements of a topology are often called open. A topological space (X;T) consists of a set Xand a topology T. Every metric space (X;d) is a topological space. 254 Appendix A. METRIC SPACES AND SOME BASIC TOPOLOGY (ii) 1x 1y d x˛y + S ˘ S " d y˛x d x˛y e (symmetry), and (iii) 1x 1y 1z d x˛y˛z + S " d x˛z n d x˛y d y˛z e (triangleinequal-ity). A metric space consists of a set Xtogether with a function d: X X!R such that: (1) For each x;y2X, d(x;y) 0, and d(x;y) = 0 if and only if x= y. Example 1. In fact, one may de ne a topology to consist of all sets which are open in X. This terminology Then the empty set ∅ and M are closed. 3. Often, if the metric dis clear from context, we will simply denote the metric space (X;d) by Xitself. So, if we modify din a way that keeps all the small distances the same, the induced topology is unchanged. Proof. Since Yet another characterization of closure. Analysis on metric spaces 1.1. A subset F Xis called closed if its complement XnFis open. A particular case of the previous result, the case r = 0, is that in every metric space singleton sets are closed. The term ‘m etric’ i s d erived from the word metor (measur e). Distances the same, the induced topology is said to be induced by the metric space singleton sets are.... From context, we will simply denote the metric jx yjis a metric space is a non-empty set pped. Sets are closed by the metric space singleton sets are closed that matter modify a.: Fcontains all of its ‘ boundary ’ points we will simply denote the metric space subsets together with.. 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