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( Topology)
Topology (Greek Topologia, ??p?????a, from topos, t?p??, "place," and logos, ?????, "study") is the branch of mathematics that studies the properties of a space that are preserved under continuous deformations. Topology grew out of geometry, but unlike geometry, topology is not concerned with metric properties such as distances between points. Instead, topology involves the study of properties that describe how a space is assembled, such as connectedness and orientability. Topology lies in the search for solution of a problem relating to the geometry of position in the true sense of the term. The word topology is used both for the area of study and for a family of sets with certain properties that are used to define a topological space, the most basic object studied in topology. Of particular importance in the study of topology are the deformations called homeomorphisms. Informally, these functions can be thought of as those that stretch space without tearing it apart or sticking distinct parts together. A more abstract notion of deformation is homotopy equivalence, which also plays a fundamental role. When the discipline was first properly founded, toward the end of the 19th century, it was called geometria situs (Latin geometry of place) and analysis situs (Latin analysis of place). From around 1925 to 1975 it was an important growth area within mathematics. Topology is a large branch of mathematics that includes many subfields. The most basic and traditional division within topology is point-set topology, which establishes the foundational aspects of topology and investigates concepts as compactness and connectedness; algebraic topology, which generally tries to measure degrees of connectivity using algebraic constructs such as homotopy groups and homology; and geometric topology, which primarily studies manifolds and their embeddings (placements) in other manifolds. Some of the most active areas, such as low dimensional topology, do not fit neatly in this division.
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Topology Subcategories
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