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Derived set (mathematics)

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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.

A subset S of a topological space is closed precisely when $S' \subseteq S$ . 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).

The set S is defined to be perfect if S = S′. Equivalently, a perfect set is a closed set with no isolated points.

Two topological spaces are homeomorphic if and only if there is a bijection from one to the other such that the derived set of the image of any subset is the image of the derived set of that subset.

The Cantor-Bendixson theorem states that any Polish space can be written as the union of a countable set and a perfect set. Because any G_δ subset of a Polish space is again a Polish space, the theorem also shows that any G_δ subset of a Polish space is the union of a countable set and a perfect set.

Because homeomorphisms can be described entirely in terms of derived sets, derived sets have been used as the primitive notion in topology. A set of points X can be equipped with an operator ^* mapping subsets of X to subsets of X, such that for any set S and any point a:

$\empty^* = \empty$
$S^{**} \subseteq S^*$
$a \in S^* \to a \in (S \setminus \{a\})^*$
$(S \cup T)^* \subseteq S^* \cup T^*$
$S \subseteq T \to S^* \subseteq T^*$

Note that given 5, 3 is equivalent to 3' below, and that 4 and 5 together are equivalent to 4' below, so we have the following equivalent axioms:

$\empty^* = \empty$
$S^{**} \subseteq S^*$

3'. $S^* = (S \setminus \{a\})^*$
4'. $\, (S \cup T)^* = S^* \cup T^*$

If we call a set S closed if $S^* \subseteq S$ , this will define a topology on the space in which ^* is the derived set operator, that is, $S^* = S' \,\!$ . If we also require that the derived set of a set consisting of a single element be empty, the resulting space will be a T₁ space.