Derived set (mathematics)

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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.

Definition

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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.

Examples

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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  [1]

Properties

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If   and   are subsets of the topological space   then the derived set has the following properties:[2]

  •  
  •   implies  
  •  
  •   implies  

A subset   of a topological space is closed precisely when  [1] that is, when   contains all its limit points. For any subset   the set   is closed and is the closure of   (that is, the set  ).[3]

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.[proof 1] In addition, if   is a T1 space, the derived set of every subset of   is closed in  [4][5]

Two subsets   and   are separated precisely when they are disjoint and each is disjoint from the other's derived set  [6]

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.[7]

A space is a T1 space if every subset consisting of a single point is closed.[8] 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.[9] It can also be shown that in a T1 space,   for any subset  [10]

A set   with   (that is,   contains no isolated points) is called dense-in-itself. A set   with   is called a perfect set.[11] Equivalently, a perfect set is a closed dense-in-itself set, or, put another way, a closed set with no isolated points. Perfect sets are particularly important in applications of the Baire category theorem.

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 Gδ subset of a Polish space is again a Polish space, the theorem also shows that any Gδ 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

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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   can be equipped with an operator   mapping subsets of   to subsets of   such that for any set   and any point  :

  1.  
  2.  
  3.   implies  
  4.  
  5.   implies  

Calling a set   closed if   will define a topology on the space in which   is the derived set operator, that is,  

Cantor–Bendixson rank

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For ordinal numbers   the  -th Cantor–Bendixson derivative of a topological space is defined by repeatedly applying the derived set operation using transfinite recursion as follows:

  •  
  •  
  •   for limit ordinals  

The transfinite sequence of Cantor–Bendixson derivatives of   is decreasing and must eventually be constant. The smallest ordinal   such that   is called the Cantor–Bendixson rank of  

This investigation into the derivation process was one of the motivations for introducing ordinal numbers by Georg Cantor.

See also

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  • Adherent point – Point that belongs to the closure of some given subset of a topological space
  • Condensation point – a stronger analog of limit point
  • Isolated point – Point of a subset S around which there are no other points of S
  • Limit point – Cluster point in a topological space

Notes

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  1. ^ a b Baker 1991, p. 41
  2. ^ Pervin 1964, p.38
  3. ^ Baker 1991, p. 42
  4. ^ Engelking 1989, p. 47
  5. ^ "General topology - Proving the derived set $E'$ is closed".
  6. ^ Pervin 1964, p. 51
  7. ^ Hocking, John G.; Young, Gail S. (1988) [1961], Topology, Dover, p. 4, ISBN 0-486-65676-4
  8. ^ Pervin 1964, p. 70
  9. ^ Kuratowski 1966, p.77
  10. ^ Kuratowski 1966, p.76
  11. ^ Pervin 1964, p. 62

Proofs

  1. ^ Proof: Assuming   is a closed subset of   which shows that   take the derived set on both sides to get   that is,   is closed in  

References

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Further reading

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