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The Metric Topology

The chapter is an extensive account of the metric topology and is a prerequisite for all the subsequent chapters. The leading sections develop the basic metric properties such as closure and interior, continuity and equivalent metrics, separation properties, product spaces, and countability axioms. This is followed by a detailed study of completeness, compactness, local compactness, and function spaces. Chapter applications include contraction mappings, continuous nowhere differentiable functions, space-filling curves, closed convex subsets of ?n, and a number of approximation results. The chapter concludes with a detailed section on orthogonal polynomials and Fourier series of continuous functions, which, together with section 3.7, provides an excellent background for Hilbert spaces. The study of sequence and function spaces in this chapter leads up gradually into Banach spaces.