## Metric SpacesSince the last century, the postulational method and an abstract point of view have played a vital role in the development of modern mathematics. The experience gained from the earlier concrete studies of analysis point to the importance of passage to the limit. The basis of this operation is the notion of distance between any two points of the line or the complex plane. The algebraic properties of underlying sets often play no role in the development of analysis; this situation naturally leads to the study of metric spaces. The abstraction not only simplifies and elucidates mathematical ideas that recur in different guises, but also helps eco- mize the intellectual effort involved in learning them. However, such an abstract approach is likely to overlook the special features of particular mathematical developments, especially those not taken into account while forming the larger picture. Hence, the study of particular mathematical developments is hard to overemphasize. The language in which a large body of ideas and results of functional analysis are expressed is that of metric spaces. The books on functional analysis seem to go over the preliminaries of this topic far too quickly. The present authors attempt to provide a leisurely approach to the theory of metric spaces. In order to ensure that the ideas take root gradually but firmly, a large number of examples and counterexamples follow each definition. Also included are several worked examples and exercises. Applications of the theory are spread out over the entire book. |

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¼ xn arbitrary arcwise connected ball S(x Cauchy sequence centre closed sets closed subset complete metric space completes the proof connected subset Consider contains continuous functions continuous mapping contraction mapping contradiction converges uniformly Corollary countable deﬁned Deﬁnition denote the set disjoint Example exists a positive exists an integer f is continuous f uniformly ffiffiffi ﬁnite finite subcover follows function defined function f hence Hint homeomorphism implies integer n0 intersection irrational jf(x Let f Let X ¼ Let X,d limit point mapping f metric induced metric space X,d Moreover natural numbers nonempty open nonempty subset open ball open cover open interval open sets open subset point of F pointwise positive integer positive number Proposition prove rational number real numbers sequence xn}n$1 Suppose Theorem totally bounded uniform convergence uniform metric uniformly continuous usual metric