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 ${\displaystyle S'}$.

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.

A subset S of a topological space is closed precisely when ${\displaystyle S'\subseteq S}$, i.e. when ${\displaystyle S}$ contains all its limit points. Two subsets S and T are separated precisely when they are disjoint and each is disjoint from the other's derived set (though the derived sets don't need to be disjoint from each other).

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