# 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 *.
*

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.

## Properties

A subset *S* of a topological space is closed precisely when , when 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).

The set *S* is defined to be a **perfect set** if . Equivalently, a perfect set is a closed set with no isolated points. Perfect sets are particularly important in applications of the Baire category theorem.

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 set that is perfect with respect to the induced topology.

## Topology in terms of derived sets

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*:

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:

Calling a set S *closed* if will define a topology on the space in which ^{*} is the derived set operator, that is, . If we also require that the derived set of a set consisting of a single element be empty, the resulting space will be a T_{1} space. In fact, 2 and 3' can fail in a space that is not T_{1}.

## Cantor–Bendixson rank

For ordinal numbers *α*, the *α*-th **Cantor–Bendixson derivative** of a topological space is defined by transfinite induction as follows:

- for limit ordinals
*λ*.

The transfinite sequence of Cantor–Bendixson derivatives of *X* must eventually be constant. The smallest ordinal *α* such that *X*^{α+1} = *X*^{α} is called the **Cantor–Bendixson rank** of *X*.

## See also

## External links

## References

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- Sierpiński, Wacław F.; translated by Krieger, C. Cecilia (1952).
*General Topology*. University of Toronto Press.