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This textbook presents the theory of Metric Spaces necessary for studying analysis beyond one real variable. Rich in examples, exercises and motivation, it provides a careful and clear exposition at a pace appropriate to the material. The book covers the main topics of metric space theory that the student of analysis is likely to need. Starting with an overview defining the principal examples of metric spaces in analysis (chapter 1), it turns to the basic theory (chapter 2) covering open and closed sets, convergence, completeness and continuity (including a treatment of continuous linear mappings). There is also a brief dive into general topology, showing how metric spaces fit into a wider theory. The following chapter is devoted to proving the completeness of the classical spaces. The text then embarks on a study of spaces with important special properties. Compact spaces, separable spaces, complete spaces and connected spaces each have a chapter devoted to them. A particular feature of the book is the occasional excursion into analysis. Examples include the Mazur-Ulam theorem, Picard's theorem on existence of solutions to ordinary differential equations, and space filling curves. This text will be useful to all undergraduate students of mathematics, especially those who require metric space concepts for topics such as multivariate analysis, differential equations, complex analysis, functional analysis, and topology. It includes a large number of exercises, varying from routine to challenging. The prerequisites are a first course in real analysis of one real variable, an acquaintance with set theory, and some experience with rigorous proofs.

Topology --- Geometry --- Mathematical analysis --- Mathematics --- analyse (wiskunde) --- wiskunde --- geometrie --- topologie --- Metric spaces. --- Mathematics. --- Espais mètrics

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Markov processes. --- Metric spaces. --- Processos de Markov --- Espais mètrics --- Espais generalitzats --- Teoria de conjunts --- Topologia --- Processos estocàstics --- Moviment brownià --- Processos de difusió --- Processos de moviment brownià --- Spaces, Metric --- Generalized spaces --- Set theory --- Topology --- Analysis, Markov --- Chains, Markov --- Markoff processes --- Markov analysis --- Markov chains --- Markov models --- Models, Markov --- Processes, Markov --- Stochastic processes

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Geometria --- Topologia --- Poliedres --- Teoria de conjunts --- Aplicacions (Matemàtica) --- Categories (Matemàtica) --- Compactificacions --- Conjunts de Borel --- Dinàmica topològica --- Espais mètrics --- Espais vectorials topològics --- Grups de transformacions --- Grups topològics --- Jocs d'estratègia (Matemàtica) --- Politops --- Teoria de l'homotopia --- Teoria de la dimensió (Topologia) --- Teoria de la dualitat (Matemàtica) --- Teoria de grafs --- Teoria dels reticles --- Topologia algebraica --- Topologia combinatòria --- Topologia diferencial --- Varietats (Matemàtica) --- Varietats topològiques --- Xarxes (Matemàtica) --- Àlgebra lineal --- Matemàtica --- Congruències (Geometria) --- Dibuix lineal --- Envolupants (Geometria) --- Esfera --- Geometria algebraica --- Geometria computacional --- Geometria conforme --- Geometria convexa --- Geometria de l'espai --- Geometria diferencial --- Geometria euclidiana --- Porismes --- Programació geomètrica --- Ràtio i proporció --- Similitud (Geometria) --- Teorema de Pitàgores --- Transformacions (Matemàtica) --- Trigonometria --- Geometria en l'art --- Geometry --- Topology

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This second of the three-volume book is targeted as a basic course in topology for undergraduate and graduate students of mathematics. It focuses on many variants of topology and its applications in modern analysis, geometry, algebra, and the theory of numbers. Offering a proper background on topology, analysis, and algebra, this volume discusses the topological groups and topological vector spaces that provide many interesting geometrical objects which relate algebra with geometry and analysis. This volume follows a systematic and comprehensive elementary approach to the topology related to manifolds, emphasizing differential topology. It further communicates the history of the emergence of the concepts leading to the development of topological groups, manifolds, and also Lie groups as mathematical topics with their motivations. This book will promote the scope, power, and active learning of the subject while covering a wide range of theories and applications in a balanced unified way.

Topology. --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Geometry --- Polyhedra --- Set theory --- Algebras, Linear --- Topologia --- Poliedres --- Teoria de conjunts --- Aplicacions (Matemàtica) --- Categories (Matemàtica) --- Compactificacions --- Conjunts de Borel --- Dinàmica topològica --- Espais mètrics --- Espais vectorials topològics --- Grups de transformacions --- Grups topològics --- Jocs d'estratègia (Matemàtica) --- Politops --- Teoria de l'homotopia --- Teoria de la dimensió (Topologia) --- Teoria de la dualitat (Matemàtica) --- Teoria de grafs --- Teoria dels reticles --- Topologia algebraica --- Topologia combinatòria --- Topologia diferencial --- Varietats (Matemàtica) --- Varietats topològiques --- Xarxes (Matemàtica) --- Àlgebra lineal --- Mathematical analysis. --- Analysis. --- 517.1 Mathematical analysis --- Mathematical analysis

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The book addresses the rigorous foundations of mathematical analysis. The first part presents a complete discussion of the fundamental topics: a review of naive set theory, the structure of real numbers, the topology of R, sequences, series, limits, differentiation and integration according to Riemann. The second part provides a more mature return to these topics: a possible axiomatization of set theory, an introduction to general topology with a particular attention to convergence in abstract spaces, a construction of the abstract Lebesgue integral in the spirit of Daniell, and the discussion of differentiation in normed linear spaces. The book can be used for graduate courses in real and abstract analysis and can also be useful as a self-study for students who begin a Ph.D. program in Analysis. The first part of the book may also be suggested as a second reading for undergraduate students with a strong interest in mathematical analysis.

Mathematical analysis. --- Analysis. --- 517.1 Mathematical analysis --- Mathematical analysis --- Anàlisi matemàtica --- Teoria de conjunts --- Agregats (Matemàtica) --- Classes (Matemàtica) --- Conjunts (Matemàtica) --- Matemàtica --- Àlgebra de Boole --- Aritmètica --- Conjunts analítics --- Conjunts convexos --- Conjunts ordenats --- Espais mètrics --- Forcing (Teoria de models) --- Funcions --- Morfismes (Matemàtica) --- Nombres ordinals --- Teoria axiomàtica de conjunts --- Teoria combinatòria de conjunts --- Teoria dels reticles --- Teoria descriptiva de conjunts --- Topologia --- Lògica matemàtica --- Àlgebra lineal --- Anàlisi combinatòria --- Anàlisi de Fourier --- Anàlisi estocàstica --- Anàlisi matemàtica no-estàndard --- Anàlisi numèrica --- Matemàtica per a enginyers --- Sèries infinites --- Teoria del potencial (Matemàtica) --- Teories no lineals --- Rutes aleatòries (Matemàtica) --- Àlgebra --- Càlcul --- Teoria de conjunts.