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The book describes some interactions of topology with other areas of mathematics and it requires only basic background. The first chapter deals with the topology of pointwise convergence and proves results of Bourgain, Fremlin, Talagrand and Rosenthal on compact sets of Baire class-1 functions. In the second chapter some topological dynamics of beta-N and its applications to combinatorial number theory are presented. The third chapter gives a proof of the Ivanovskii-Kuzminov-Vilenkin theorem that compact groups are dyadic. The last chapter presents Marjanovic's classification of hyperspaces of compact metric zerodimensional spaces.

Analytical topology --- Topology --- Geometry --- Mathematical Theory --- Mathematics --- Physical Sciences & Mathematics --- Topologie --- Topology. --- Topological groups. --- Lie groups. --- Functions of real variables. --- Topological Groups, Lie Groups. --- Real Functions. --- Real variables --- Functions of complex variables --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Groups, Topological --- Continuous groups --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Polyhedra --- Set theory --- Algebras, Linear

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Ramsey theory is a fast-growing area of combinatorics with deep connections to other fields of mathematics such as topological dynamics, ergodic theory, mathematical logic, and algebra. The area of Ramsey theory dealing with Ramsey-type phenomena in higher dimensions is particularly useful. Introduction to Ramsey Spaces presents in a systematic way a method for building higher-dimensional Ramsey spaces from basic one-dimensional principles. It is the first book-length treatment of this area of Ramsey theory, and emphasizes applications for related and surrounding fields of mathematics, such as set theory, combinatorics, real and functional analysis, and topology. In order to facilitate accessibility, the book gives the method in its axiomatic form with examples that cover many important parts of Ramsey theory both finite and infinite. An exciting new direction for combinatorics, this book will interest graduate students and researchers working in mathematical subdisciplines requiring the mastery and practice of high-dimensional Ramsey theory.

Algebraic spaces. --- Ramsey theory. --- Ramsey theory --- Algebraic spaces --- Mathematics --- Algebra --- Physical Sciences & Mathematics --- Spaces, Algebraic --- Geometry, Algebraic --- Combinatorial analysis --- Graph theory --- Analytic set. --- Axiom of choice. --- Baire category theorem. --- Baire space. --- Banach space. --- Bijection. --- Binary relation. --- Boolean prime ideal theorem. --- Borel equivalence relation. --- Borel measure. --- Borel set. --- C0. --- Cantor cube. --- Cantor set. --- Cantor space. --- Cardinality. --- Characteristic function (probability theory). --- Characterization (mathematics). --- Combinatorics. --- Compact space. --- Compactification (mathematics). --- Complete metric space. --- Completely metrizable space. --- Constructible universe. --- Continuous function (set theory). --- Continuous function. --- Corollary. --- Countable set. --- Counterexample. --- Decision problem. --- Dense set. --- Diagonalization. --- Dimension (vector space). --- Dimension. --- Discrete space. --- Disjoint sets. --- Dual space. --- Embedding. --- Equation. --- Equivalence relation. --- Existential quantification. --- Family of sets. --- Forcing (mathematics). --- Forcing (recursion theory). --- Gap theorem. --- Geometry. --- Ideal (ring theory). --- Infinite product. --- Lebesgue measure. --- Limit point. --- Lipschitz continuity. --- Mathematical induction. --- Mathematical problem. --- Mathematics. --- Metric space. --- Metrization theorem. --- Monotonic function. --- Natural number. --- Natural topology. --- Neighbourhood (mathematics). --- Null set. --- Open set. --- Order type. --- Partial function. --- Partially ordered set. --- Peano axioms. --- Point at infinity. --- Pointwise. --- Polish space. --- Probability measure. --- Product measure. --- Product topology. --- Property of Baire. --- Ramsey's theorem. --- Right inverse. --- Scalar multiplication. --- Schauder basis. --- Semigroup. --- Sequence. --- Sequential space. --- Set (mathematics). --- Set theory. --- Sperner family. --- Subsequence. --- Subset. --- Subspace topology. --- Support function. --- Symmetric difference. --- Theorem. --- Topological dynamics. --- Topological group. --- Topological space. --- Topology. --- Tree (data structure). --- Unit interval. --- Unit sphere. --- Variable (mathematics). --- Well-order. --- Zorn's lemma.

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Number theory --- Discrete mathematics --- discrete wiskunde --- getallenleer

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Mathematical logic --- Ordered algebraic structures --- Discrete mathematics --- algebra --- discrete wiskunde --- wiskunde --- logica

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The analysis of the characteristics of walks on ordinals is a powerful new technique for building mathematical structures, developed by the author over the last twenty years. This is the first book-length exposition of this method. Particular emphasis is placed on applications which are presented in a unified and comprehensive manner and which stretch across several areas of mathematics such as set theory, combinatorics, general topology, functional analysis, and general algebra. The intended audience for this book are graduate students and researchers working in these areas interested in mastering and applying these methods.

Number theory --- Discrete mathematics --- discrete wiskunde --- getallenleer