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This is the second of a two-volume work intended to function as a textbook well as a reference work for economic for graduate students in economics, as scholars who are either working in theory, or who have a strong interest in economic theory. While it is not necessary that a student read the first volume before tackling this one, it may make things easier to have done so. In any case, the student undertaking a serious study of this volume should be familiar with the theories of continuity, convergence and convexity in Euclidean space, and have had a fairly sophisticated semester's work in Linear Algebra. While I have set forth my reasons for writing these volumes in the preface to Volume 1 of this work, it is perhaps in order to repeat that explanation here. I have undertaken this project for three principal reasons. In the first place, I have collected a number of results which are frequently useful in economics, but for which exact statements and proofs are rather difficult to find; for example, a number of results on convex sets and their separation by hyperplanes, some results on correspondences, and some results concerning support functions and their duals. Secondly, while the mathematical top ics taken up in these two volumes are generally taught somewhere in the mathematics curriculum, they are never (insofar as I am aware) done in a two-course sequence as they are arranged here.
Content Level »Graduate
Keywords »Connected space - Economic Theory - Mathematical Methods - Mathematics - Mathematik - Wirtschaftstheorie - mathematische Methoden - set
An Introduction to Topology.- Basic Concepts.- Closed Sets and Closures.- Topological Bases.- Continuous Functions.- Metric Spaces.- Complete Metric Spaces.- Nets and Convergence.- Additional Topics in Topology.- Relative and Product Topologies.- Compactness.- Hausdorff and Normal Spaces.- Compact Metric Spaces.- Connected Spaces.- Paracompactness and Partitions of Unity.- Correspondences.- Preliminary Considerations.- Hemi-Continuous Correspondences.- Correspondences Defined by Functions.- Closed Correspondences.- The Domain and Range of Correspondences.- Compositions of Correspondences.- Operations with Correspondences.- Correspondences into Metric Spaces.- Open Correspondences and Open Sections.- Banach Spaces.- Preliminaries.- An Introduction to Banach Spaces.- Bounded Linear Mappings.- Some Fundamental Theorems.- Dual Spaces.- Topological Vector Spaces.- Introduction.- Continuous Functions and Convex Sets.- Separation Theorems.- Equilibrium Models in Hilbert Space.- Locally Convex Spaces.- Correspondences.- Selection and Fixed Point Theorems.- Maximum Theorems.- Sperner`s Lemma and the K-K-M Theorem.- Fixed Point Theorems.- Selection Theorems.- Equilibrium in an 'Abstract Economy'.