Let p ∈ M and r ≥ 0. Often, if the metric dis clear from context, we will simply denote the metric space (X;d) by Xitself. The elements of a topology are often called open. Let M be an arbitrary metric space. 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. Let (X;T ) be a topological space. Proof. This distance function is known as the metric. Metric Spaces, Topological Spaces, and Compactness Proposition A.6. 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. A topological space (X;T) consists of a set Xand a topology T. Every metric space (X;d) is 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. 3. In fact, one may de ne a topology to consist of all sets which are open in X. De nition (Metric space). Picture for a closed set: Fcontains all of its ‘boundary’ points. The term ‘m etric’ i s d erived from the word metor (measur e). Pick xn 2 Kn. Fix then Take . 78 CHAPTER 3. A particular case of the previous result, the case r = 0, is that in every metric space singleton sets are closed. Topology of metric space Metric Spaces Page 3 . 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. Let Xbe a compact metric space. 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 . 1. 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. Remark 1.1.11. Prove that there is a topology on Xconsisting of the co nite subsets together with ;. A subset U of X is co nite if XnU is nite. This terminology A subset F Xis called closed if its complement XnFis open. The closure of a set is defined as Theorem. 1 THE TOPOLOGY OF METRIC SPACES 3 1. A metric space is a set Xtogether with a metric don it, and we will use the notation (X;d) for a metric space. (Alternative characterization of the closure). Then the closed ball of center p, radius r; that is, the set {q ∈ M: d(q,p) ≤ r} is closed. 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). Since Yet another characterization of closure. 2. So, if we modify din a way that keeps all the small distances the same, the induced topology is unchanged. Let M be an arbitrary metric space. y. More If each Kn 6= ;, then T n Kn 6= ;. Assume K1 ˙ K2 ˙ K3 ˙ form a decreasing sequence of closed subsets of X. 5.Let X be a set. This particular topology is said to be induced by the metric. a metric space, it is only the small distances that matter. De nitions, and open sets. 254 Appendix A. Analysis on metric spaces 1.1. If (A) holds, (xn) has a convergent subsequence, xn k! The set of real numbers R with the function d(x;y) = jx yjis a metric space. Proof. Note that iff If then so Thus On the other hand, let . Example 1. iff ( is a limit point of ). Then the empty set ∅ and M are closed. A metric space is a non-empty set equi pped with structure determined by a well-defin ed notion of distan ce. Said to be induced by the metric dis clear from context, we will denote. Subsequence, xn k Fcontains all of its ‘ boundary ’ points s!, Topological Spaces, and Compactness Proposition A.6 Spaces, Topological Spaces, and Compactness Proposition A.6 if ( )... 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