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Separation Relation
A separation relation may refer to * Betweenness relation * Point-pair separation in a cycle * Separation axioms In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. These are sometimes ... in point-set topology, or * Arm's length principle {{Disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Betweenness Relation
Betweenness is a noun derived from the proposition between. It may refer to: * The ternary relation of intermediacy or betweenness, a feature of ordered geometry. * Betweenness problem - an algorithmic problem. The input is a collection of ordered triples of items; the task is to decide whether there is a single total order such that such that, for each of the given triples, the middle item in the triple appears in the output somewhere between the other two items. * Betweenness centrality - a measure of centrality in a graph, based on shortest paths. The betweenness centrality of a vertex is the number of shortest paths that pass through the vertex. * Metric betweenness - given a metric ''d'', a point ''y'' is said to be ''between'' ''x'' and ''z'' if all three points are distinct, and d(x,y)+d(y,z)=d(x,z). See convex metric space. See also * Between (other) * In Between (other) {{Disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Point-pair Separation
In a cyclic order, such as the real projective line, two pairs of points separate each other when they occur alternately in the order. Thus the ordering ''a b c d'' of four points has (''a,c'') and (''b,d'') as separating pairs. This point-pair separation is an invariant of projectivities of the line. Concept The concept was described by G. B. Halsted at the outset of his ''Synthetic Projective Geometry'': Given any pair of points on a projective line, they separate a third point from its harmonic conjugate. A pair of lines in a pencil separates another pair when a transversal crosses the pairs in separated points. The point-pair separation of points was written AC//BD by H. S. M. Coxeter in his textbook ''The Real Projective Plane''. Application The relation may be used in showing the real projective plane is a complete space. The axiom of continuity used is "Every monotonic sequence of points has a limit." The point-pair separation is used to provide definitions: * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separation Axioms
In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. These are sometimes called ''Tychonoff separation axioms'', after Andrey Tychonoff. The separation axioms are not fundamental axioms like those of Zermelo–Fraenkel set theory, set theory, but rather defining properties which may be specified to distinguish certain types of topological spaces. The separation axioms are denoted with the letter "T" after the German language, German ''Trennungsaxiom'' ("separation axiom"), and increasing numerical subscripts denote stronger and stronger properties. The precise definitions of the history of the separation axioms, separation axioms have varied over time. Especially in older literature, different authors might have different definitions of each condition. Preliminary definitions Before we define the separation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
