Editing Derived set (mathematics) (section)

==Properties==

Let <math>X</math> denote a topological space in what follows.

If <math>A</math> and <math>B</math> are subsets of <math>X,</math> the derived set has the following properties:<ref>{{harvnb|Pervin|1964|loc=p.38}}</ref>
* <math>\varnothing' = \varnothing</math>
* <math>a \in A'</math> implies <math>a \in (A \setminus \{a\})'</math>
* <math>(A \cup B)' = A' \cup B'</math>
* <math>A \subseteq B</math> implies <math>A' \subseteq B'</math>

A set <math>S\subseteq X</math> is [[closed set|closed]] precisely when <math>S' \subseteq S,</math><ref name=Baker41 /> that is, when <math>S</math> contains all its limit points. For any <math>S\subseteq X,</math> the set <math>S \cup S'</math> is closed and is the [[Closure (topology)|closure]] of <math>S</math> (that is, the set <math>\overline{S}</math>).<ref>{{harvnb|Baker|1991|loc=p. 42}}</ref>

===Closedness of derived sets===

The derived set of a set need not be closed in general.  For example, if <math>X = \{a, b\}</math> with the [[indiscrete topology]], the set <math>S = \{a\}</math> has derived set <math>S' = \{b\},</math> which is not closed in <math>X.</math>  But the derived set of a closed set is always closed.<ref group=proof>''Proof:'' Assuming <math>S</math> is a closed subset of <math>X,</math> which shows that <math>S' \subseteq S,</math> take the derived set on both sides to get <math>S'' \subseteq S';</math> that is, <math>S'</math> is closed in <math>X.</math></ref>

For a point <math>x\in X,</math> the derived set of the singleton <math>\{x\}</math> is the set <math>\{x\}'=\overline{\{x\}}\setminus\{x\},</math> consisting of the points in the closure of <math>\{x\}</math> and different from <math>x.</math>
A space <math>X</math> is called a '''T<sub>D</sub> space'''<ref name="aull-thron">{{cite journal |last1=Aull |first1=C. E. |last2=Thron |first2=W. J. |title=Separation axioms between T0 and T1 |journal=Nederl. Akad. Wetensch. Proc. Ser. A |date=1962 |volume=65 |pages=26–37 |doi=10.1016/S1385-7258(62)50003-6 |url=https://core.ac.uk/download/pdf/82702431.pdf |zbl=0108.35402}}Definition 3.1</ref> if the derived set of every singleton in <math>X</math> is closed; that is, if <math>\overline{\{x\}}\setminus\{x\}</math> is closed for every <math>x\in X;</math> in other words, if every point <math>x</math> is isolated in <math>\overline{\{x\}}.</math>  A space <math>X</math> has the property that <math>S'</math> is closed for all sets <math>S\subseteq X</math> if and only if it is a T<sub>D</sub> space.{{sfn|Aull|Thron|1962|loc=Theorem 5.1}}

Every T<sub>D</sub> space is a [[T0 space|T<sub>0</sub> space]].<ref name="jgl">{{cite web|last1=Goubault-Larrecq |first1=Jean |title=TD spaces |url=https://projects.lsv.ens-cachan.fr/topology/?page_id=2626 |website=Non-Hausdorff Topology and Domain Theory}}</ref>

Every [[T1 space|T<sub>1</sub> space]] is a T<sub>D</sub> space,<ref name="jgl"/> since every singleton is closed, hence
<math>\{x\}'=\overline{\{x\}}\setminus\{x\}=\varnothing,</math> which is closed.
Consequently, in a T<sub>1</sub> space, the derived set of any set is closed.<ref>{{harvnb|Engelking|1989|loc=p. 47}}</ref><ref>{{Cite web|url=https://math.stackexchange.com/a/940849/52912|title=Proving the derived set E' is closed}}</ref>

The relation between these properties can be summarized as
:<math>T_1\implies T_D\implies T_0.</math>

The implications are not reversible.  For example, the [[Sierpiński space]] is T<sub>D</sub> and not T<sub>1</sub>.
And the [[right order topology]] on <math>\R</math> is T<sub>0</sub> and not T<sub>D</sub>.

===More properties===

Two subsets <math>S</math> and <math>T</math> are [[Separated sets|separated]] precisely when they are [[Disjoint sets|disjoint]] and each is disjoint from the other's derived set <math display=inline>S' \cap T = \varnothing = T' \cap S.</math><ref>{{harvnb|Pervin|1964|loc=p. 51}}</ref>

A [[bijection]] between two topological spaces is a [[homeomorphism]] if and only if the derived set of the [[image (mathematics)|image]] (in the second space) of any subset of the first space is the image of the derived set of that subset.<ref>{{citation|first1=John G.|last1=Hocking|first2=Gail S.|last2=Young|title=Topology|year=1988|orig-date=1961|publisher=Dover|isbn=0-486-65676-4|page=[https://archive.org/details/topology00hock_0/page/4 4]|url=https://archive.org/details/topology00hock_0/page/4}}</ref>

In a T<sub>1</sub> space, the derived set of any finite set is empty and furthermore,
<math display=block>(S - \{p\})' = S' = (S \cup \{p\})',</math>
for any subset <math>S</math> and any point <math>p</math> 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.<ref>{{harvnb|Kuratowski|1966|loc=p.77}}</ref>

A set <math>S</math> with <math>S \subseteq S'</math> (that is, <math>S</math> contains no [[isolated point]]s) is called [[dense-in-itself]]. A set <math>S</math> with <math>S = S'</math> is called a [[perfect set]].<ref>{{harvnb|Pervin|1964|loc=p. 62}}</ref>  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 [[G-delta set|G<sub>δ</sub>]] subset of a Polish space is again a Polish space, the theorem also shows that any G<sub>δ</sub> subset of a Polish space is the union of a countable set and a set that is perfect with respect to the [[induced topology]].