Prove that there is a topology on Xconsisting of the co nite subsets together with ;. Proof. 1 THE TOPOLOGY OF METRIC SPACES 3 1. a metric space, it is only the small distances that matter. 2. Pick xn 2 Kn. 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. Fix then Take . 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. 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. The elements of a topology are often called open. Metric Spaces, Topological Spaces, and Compactness Proposition A.6. A subset F Xis called closed if its complement XnFis open. Let p ∈ M and r ≥ 0. Then the closed ball of center p, radius r; that is, the set {q ∈ M: d(q,p) ≤ r} is closed. Let M be an arbitrary metric space. y. In fact, one may de ne a topology to consist of all sets which are open in X. A metric space is a set Xtogether with a metric don it, and we will use the notation (X;d) for a metric space. 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). Let (X;T ) be a topological space. Topology of metric space Metric Spaces Page 3 . Let Xbe a compact metric space. If (A) holds, (xn) has a convergent subsequence, xn k! 3. More Example 1. This terminology 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. 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. Picture for a closed set: Fcontains all of its ‘boundary’ points. 254 Appendix A. Proof. So, if we modify din a way that keeps all the small distances the same, the induced topology is unchanged. Since Yet another characterization of closure. If each Kn 6= ;, then T n Kn 6= ;. 78 CHAPTER 3. (Alternative characterization of the closure). The set of real numbers R with the function d(x;y) = jx yjis a metric space. This distance function is known as the metric. Then the empty set ∅ and M are closed. This particular topology is said to be induced by the metric. iff ( is a limit point of ). The term ‘m etric’ i s d erived from the word metor (measur e). A subset U of X is co nite if XnU is nite. Let M be an arbitrary 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 . Remark 1.1.11. 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