Derived set (mathematics)

In mathematics, more specifically in point-set topology, the derived set of a subset of a topological space is the set of all limit points of It is usually denoted by 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. The derived set of a subset of a topological space denoted by is the set of all points that are limit points of that is, points such that every neighbourhood of contains a point of other than itself. If is endowed with its usual Euclidean topology then the derived set of the half-open interval is the closed interval Consider with the topology (open sets) consisting of the empty set and any subset of that contains 1. The derived set of is If and are subsets of the topological space then the derived set has the following properties: implies implies A subset of a topological space is closed precisely when that is, when contains all its limit points. For any subset the set is closed and is the closure of (that is, the set ). The derived set of a subset of a space need not be closed in general. For example, if with the trivial topology, the set has derived set which is not closed in But the derived set of a closed set is always closed. In addition, if is a T1 space, the derived set of every subset of is closed in Two subsets and are separated precisely when they are disjoint and each is disjoint from the other's derived set A bijection between two topological spaces is a homeomorphism if and only if the derived set of the image (in the second space) of any subset of the first space is the image of the derived set of that subset. A space is a T1 space if every subset consisting of a single point is closed. In a T1 space, the derived set of a set consisting of a single element is empty (Example 2 above is not a T1 space). It follows that in T1 spaces, the derived set of any finite set is empty and furthermore, for any subset and any point of the space. In other words, the derived set is not changed by adding to or removing from the given set a finite number of points.