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Coherent (topology)
In topology, a coherent topology is a topology that is uniquely determined by a family of Subspace topology, subspaces. Loosely speaking, a topological space is coherent with a family of subspaces if it is a ''topological union'' of those subspaces. It is also sometimes called the weak topology generated by the family of subspaces, a notion that is quite different from the notion of a weak topology generated by a set of maps. Definition Let X be a topological space and let C = \left\ be a indexed family, family of subsets of X, each with its induced subspace topology. (Typically C will be a Cover (topology), cover of X.) Then X is said to be coherent with C (or determined by C)X is also said to have the weak topology generated by C. This is a potentially confusing name since the adjectives and are used with opposite meanings by different authors. In modern usage the term is synonymous with initial topology and is synonymous with final topology. It is the final topology that i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such as Stretch factor, stretching, Torsion (mechanics), twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a Set (mathematics), set endowed with a structure, called a ''Topology (structure), topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of List of continuity-related mathematical topics, continuity. Euclidean spaces, and, more generally, metric spaces are examples of topological spaces, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and Homotopy, homotopies. A property that is invariant under such deformations is a to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |