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Cantor-Bendixson Rank
In mathematics, more specifically in point-set topology, the derived set of a subset S of a topological space is the set of all limit points of S. It is usually denoted by S'. The concept was first introduced by Georg Cantor in 1872 and he developed set theory in large part to study derived sets on the real line. Definition The derived set of a Set (mathematics), subset S of a topological space X, denoted by S', is the set of all points x \in X that are limit points of S, that is, points x such that every neighbourhood (mathematics), neighbourhood of x contains a point of S other than x itself. Examples If \Reals is endowed with its usual Euclidean topology then the derived set of the half-open interval [0, 1) is the closed interval [0, 1]. Consider \Reals with the Topology (structure), topology (open sets) consisting of the empty set and any subset of \Reals that contains 1. The derived set of A := \ is A' = \Reals \setminus \. Properties Let X denote a topological space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Point-set Topology
In mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. The fundamental concepts in point-set topology are ''continuity'', ''compactness'', and ''connectedness'': * Continuous functions, intuitively, take nearby points to nearby points. * Compact sets are those that can be covered by finitely many sets of arbitrarily small size. * Connected sets are sets that cannot be divided into two pieces that are far apart. The terms 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using the concept of open sets. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a ''topology''. A set with a topology i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
