Introduction to Metric and Topological Spaces
One of the ways in which topology has influenced other branches of mathematics in the past few decades is by putting the study of continuity and convergence into a general setting. This book introduces metric and topological spaces by describing some of that influence. The aim is to move gradually from familiar real analysis to abstract topological spaces. The book is aimed primarily at the second-year mathematics student, and numerous exercises are included.
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THE HAUSDORFF CONDITION
COMPLETE METRIC SPACES
CRITERIA FOR COMPACTNESS
GUIDE TO EXERCISES
applications basis called Cauchy sequence Chapter choose closed collection compact compact metric space complete condition connected consider constant contains continuity of f continuous map contradiction converges COROLLARY correspondence defined definition denote dense discrete disjoint distance easy equicontinuous equivalence Euclidean example Exercise exists f is continuous finite finite subcover fixed follows give given graph Hausdorff Hence holds homeomorphic inductively inequality integer intersection interval intuitive least Lemma limit Lipschitz means metric space non-empty open ball open cover open set particular partitions Proof Proposition Prove rational reader real numbers real-valued functions refer relation result satisfying sense similar Similarly subsequence subset subspace sufficiently Suppose theorem tion topological space totally bounded true uniform union unique upper bound values variables