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*Mathematics | Program Guide: Topology
A guide to online Mathematics research for APUS faculty and students.
A Guide to Topology, an introduction to basic topology, covers point-set topology, Moore-Smith convergence and function spaces. It treats continuity, compactness, the separation axioms, connectedness, completeness, the relative topology, the quotient topology, the product topology, and all the other fundamental ideas of the subject. The book is filled with examples and illustrations. Students studying for exams will find this book to be a concise, focused and informative resource.
The interplay between algebra and geometry is a beautiful (and fun!) area of mathematical investigation. Recent advances in computing and algorithms make it possible to tackle many classical problems in a down-to-earth and concrete fashion. This opens wonderful new vistas and allows us to pose, study and solve problems that were previously out of reach. The objective of this book is to bring advanced algebra to life with lots of examples.
"This clear and winning little book, for readers willing to come to genuine grips with the idea of a mathematical proof, presents topology ... as mathematicians see it. One cannot any longer doubt that a single stroke of a knife exists that divides any irregular ham sandwich so that the ham and both bread slices can be shared with perfect fairness by two consumers." — Scientific American
This edited collection bridges the foundations and practice of constructive mathematics and focusses on the contrast between the theoretical developments, which have been most useful for computer science (eg constructive set and type theories), and more specific efforts on constructive analysis, algebra and topology.
Geometry provides a whole range of views on the universe, serving as the inspiration, technical toolkit and ultimate goal for many branches of mathematics and physics. This book introduces the ideas of geometry, and includes a generous supply of simple explanations and examples. With minimal prerequisites, the book provides a first glimpse of many research topics in modern algebra, geometry and theoretical physics. There are copious illustrations, and each chapter ends with a wide supply of exercises. Further teaching material is available for teachers via the web, including assignable problem sheets with solutions.
This volume is a collection of surveys of research problems in topology and its applications. The topics covered include general topology, set-theoretic topology, continuum theory, topological algebra, dynamical systems, computational topology and functional analysis.
This book provides exposition of the subject both in its general and algebraic aspects. It deals with the notions of topological spaces, compactness, connectedness, completeness including metrizability and compactification, algebraic aspects of topological spaces through homotopy groups and homology groups. It begins with the basic notions of topological spaces but soon going beyond them reaches the domain of algebra through the notions of homotopy, homology and cohomology.
YouTube video including: the definition of a topology, open and closed sets, and the basis for a topology. Examples of topologies, including the trivial topology, the discrete topology, particular point / excluded point topologies.
"Interactive Real Analysis" is an online, interactive textbook for Real Analysis or Advanced Calculus. It deals with sets, sequences, series, continuity, differentiability, integrability, topology, etc.