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Topology: Spaces
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Topology (Greek [T]opos, "place," and logos, "study") is a branch of mathematics that is an extension of geometry. Topology begins with a consideration of the nature of space, investigating both its fine structure and its global structure. Topology builds on set theory, considering both sets of points and families of sets.
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Topology is the study of how spaces are organized, how the objects are structured in terms of position. It ... studies how spaces are connected. It is divided into [A]lgebraic topology, differential topology and geometric topology.
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Topology concerns geometric concepts such as distance, connectedness, and continuity, but in a fluid, flexible way. This can be contrasted with classical geometry which focuses on the geometry of rigid properties. The subject of topology can be developed in a beautifully abstract approach that has great simplicity, great generality, and yet great power. It has important applications to a host of fields, including cosmology (the shape of space), robotics, chaos, knot theory, and even internet searching.
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See 55: Algebraic Topology for the definitions, and computations, and applications of fundamental groups, homotopy groups, homology and cohomology. This includes topics in homotopy theory -- studies of spaces in the homotopy category -- whether or not they involve algebraic invariants.
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Topology is concerned with the intrinsic properties of shapes of spaces. One class of spaces which plays a central role in mathematics, and whose topology is extensively studied, are the n dimensional manifolds. These are spaces which locally look like Euclidean n-dimensional space.
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