Denseinitself
In general topology, a subset A of a topological space is said to be denseinitself or crowded if A has no isolated point. Equivalently, A is denseinitself if every point of A is a limit point of A. Thus A is denseinitself if and only if A\subseteq A', where A' is the derived set of A. A denseinitself 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 unrelated to ''denseinitself''. This can sometimes be confusing, as "X is dense in X" (always true) is not the same as "X is denseinitself" (no isolated point). Examples A simple example of a set that is denseinitself 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 denseinitself 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 closed be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Derived Set (mathematics)
In mathematics, more specifically in pointset 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. Examples If \mathbb is endowed with its usual Euclidean topology then the derived set of the halfopen interval , 1) is the closed interval [0,1 Consider \mathbb with the Topology (structure)">topology (open sets) consisting of the empty set and any subset of \mathbb that contains 1. The derived set of A := \ is A' = \mathbb \setminus \. Properties If A and B are subsets of the topological space \left(X, \mathcal\right), then the derived set has the following properties: * \varnothing' = \varnothing * a \in A' \implies a \in (A \setminus \)' * (A \cup B)' = A' \cup B' * A \subseteq B \implies A' \subseteq B' A subset S of a topological space is closed precis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 