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Topology
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Topology is the mathematical study of those properties that are preserved through continuous deformations of objects. A circle is topologically equivalent to an ellipse, a sphere is equivalent to a cube, and a coffee cup to a donut. Topology began with the study of curves, surfaces, and other objects in the plane and three-space. It can be used to abstract the inherent connectivity of objects while ignoring their detailed form. The "objects" of topology are formally defined as topological spaces. If two spaces have the same topological properties, they are said to be homeomorphic; if one can be continuously deformed into the other, they are said to be homotopy equivalent.
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Topology is a branch of pure mathematics that deals with the abstract relationships found in geometry and analysis. The word "topology" means the study of surfaces. Topologists attempt to understand shape and space without an explicit measure of distance or size. You may hear topology described as "rubber sheet" geometry. That is, if geometric figures are drawn on a rubber sheet, then stretched and contracted, their topological properties do not change. To a topologist, there is no difference among a large square, a small disk, and a cone.
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Topology applies GIS behaviors to spatial data. Topology enables GIS software to answer questions such as adjacency, connectivity, proximity, and coincidence. In ArcGIS, a topology provides a powerful and flexible way for users to specify the rules for establishing and maintaining the quality and integrity of your spatial data. You want to be able to know, for example, that all your parcel polygons completely form closed rings, they don't overlap one another, and there are no gaps between parcels. You can ... use topology to validate the spatial relationships between feature classes. For example, the lot lines in your parcel data model must share coincident geometry with the parcel boundaries.
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Topology ... refers to a particular mathematical object studied in this area. In this sense, a [T]opology is a family of open sets which contains the empty set and the entire space. If a family of sets is in the topology, then its union must be in the topology. If a finite family of sets is in the topology, then its intersection must be in the topology. A set equipped with a topology is called a topological space. The remainder of this article deals with the branch of mathematics known as topology.
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Topology is increasingly recognised as one of Australia's leading new music ensembles. Their energetic, full-blooded sound belies their compact instrumentation. Since forming in 1997, Topology has built a solid audience, and regularly performs to sold-out houses around Australia. The group's concerts are broadcast nationally by the ABC.
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Topology is concerned with those properties of geometric figures that are invariant under continuous transformations. A continuous transformation... called a topological transformation or homeomorphism, is a one-to-one correspondence between the points of one figure and the points of another figure such that points that are arbitrarily close on one figure are transformed into points that are also arbitrarily close on the other figure. Figures that are related in this way are said to be topologically equivalent. If a figure is transformed into an equivalent figure by bending, stretching, etc., the change is a special type of topological transformation called a continuous deformation. Two figures (e.g, certain types of knots) may be topologically equivalent, however, without being changeable into one another by a continuous deformation.
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