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Dense-in-itself
In general topology, a subset A of a topological space is said to be dense-in-itself or crowded if A has no isolated point. Equivalently, A is dense-in-itself if every point of A is a limit point of A. Thus A is dense-in-itself if and only if A\subseteq A', where A' is the derived set of A. A dense-in-itself closed set is called a perfect set. (In other words, a perfect set is a closed set without isolated point.) The notion of dense set is distinct from ''dense-in-itself''. This can sometimes be confusing, as "''X'' is dense in ''X''" (always true) is not the same as "''X'' is dense-in-itself" (no isolated point). Examples A simple example of a set that is dense-in-itself but not closed (and hence not a perfect set) is the set of irrational numbers (considered as a subset of the real numbers). This set is dense-in-itself because every neighborhood of an irrational number x contains at least one other irrational number y \neq x. On the other hand, the set of irrationals is not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Derived Set (mathematics)
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 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 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 , 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 in what follows. If A and B are subsets of X, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Isolated Point
In mathematics, a point is called an isolated point of a subset (in a topological space ) if is an element of and there exists a neighborhood of that does not contain any other points of . This is equivalent to saying that the singleton is an open set in the topological space (considered as a subspace of ). Another equivalent formulation is: an element of is an isolated point of if and only if it is not a limit point of . If the space is a metric space, for example a Euclidean space, then an element of is an isolated point of if there exists an open ball around that contains only finitely many elements of . A point set that is made up only of isolated points is called a discrete set or discrete point set (see also discrete space). Related notions Any discrete subset of Euclidean space must be countable, since the isolation of each of its points together with the fact that rationals are dense in the reals means that the points of may be mapped injective ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Dense Set
In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation). Formally, A is dense in X if the smallest closed subset of X containing A is X itself. The of a topological space X is the least cardinality of a dense subset of X. Definition A subset A of a topological space X is said to be a of X if any of the following equivalent conditions are satisfied: The smallest closed subset of X containing A is X itself. The closure of A in X is equal to X. That is, \operatorname_X A = X. The interior of the complement of A is empty. That is, \operatorname_X (X \setminus A) = \varnothing. Every point in X eith ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Glossary Of Topology
This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology. For a list of terms specific to algebraic topology, see Glossary of algebraic topology. All spaces in this glossary are assumed to be topological spaces unless stated otherwise. A ;Absolutely closed: See ''H-closed'' ;Accessible: See T_1. ;Accumulation point: See limit point. ;Alexandrov topology: The topology of a space ''X'' is an Alexandrov topology (or is finitely generated) if arbitrary intersections of open sets in ''X'' are open, or equivalently, if arbitrary unions of closed sets are closed, or, again equivalently, if the open sets are the upper sets of a poset. ;Almost discrete: A space is almost discrete if every open set is closed (hence cl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |