Editing Derived set (mathematics) (section)

===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>.