First Course In Topology (eBook, ePUB)
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Topology is the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects. Tearing, however, is not allowed. A circle is topologically equivalent to an ellipse (into which it can be deformed by stretching) and a sphere is equivalent to an ellipsoid. Similarly, the set of all possible positions of the hour hand of a clock is topologically equivalent to a circle (i.e., a one-dimensional closed curve with no intersections that can be embedded in two-dimensional space), the set of all possible positions of the hour and minute hands taken to...
Topology is the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects. Tearing, however, is not allowed. A circle is topologically equivalent to an ellipse (into which it can be deformed by stretching) and a sphere is equivalent to an ellipsoid. Similarly, the set of all possible positions of the hour hand of a clock is topologically equivalent to a circle (i.e., a one-dimensional closed curve with no intersections that can be embedded in two-dimensional space), the set of all possible positions of the hour and minute hands taken together is topologically equivalent to the surface of a torus, and the set of all possible positions of the hour, minute, and second hands taken together are topologically equivalent to a three-dimensional object. Topology can be used to abstract the inherent connectivity of objects while ignoring their detailed form. This book on topology provides in depth coverage of both general topology and algebraic topology. It includes many examples and figures. It will be highly beneficial for anyone needing a basic, thorough, introduction to general and algebraic topology and its applications. Contents: Topology Basics; General Topology; Topological Space; Distinction Between Geometry and Topology; Metrization Theorem; Topological Ring; Borromean Rings; Real Projective Plane.
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