View source for Index of properties

==Index==
Note:
* Things are indexed by the ''adjective in the property'', for example: {{Sigma|finite}} is under "finite".
* The specific case contains extra information, so {{Sigma|finite}} is under finite, but specifically {{Sigma|finite}}
* The word "under" is ignored in the index
{| class="wikitable" border="1"
|-
! Adjective
! Specific case
! Index
! Description
|-
! rowspan="8" | Closed
| ''(general)''
| CLOSED
| To say something is ''closed under'' means one cannot leave it through the stated property, eg "the integers are ''closed under'' addition
|-
| {{M|\backslash}}-closed<ref name="PTACC">Probability Theory - A comprehensive course - Second Edition - Achim Klenke</ref>
| CLOSED_backslash
| To say {{M|\mathcal{A} }} is {{M|\backslash}}-closed uses {{M|\backslash}} to denote [[Set subtraction|set subtraction]]<ref group="Note">This is because {{M|-}}-closed is not a good way to write this</ref>, this means <math>\forall A,B\in\mathcal{A}[A-B\in\mathcal{A}]</math>
|-
| {{M|\cap}}-closed<ref name="PTACC"/>
| CLOSED_cap
| If {{M|\mathcal{A} }} is ''{{M|\cap}}-closed'' then <math>\forall A,B\in\mathcal{A}[A\cap B\in\mathcal{A}]</math> - {{M|\mathcal{A} }} is closed under finite [[Intersection|intersection]]
|-
| {{Sigma|{{M|\cap}}-closed}}<ref name="PTACC"/>
| CLOSED_cap_sigma
| closed under [[Countably infinite|countably infinite]] intersection. <math>\forall (A_n)_{n=1}^\infty\subseteq\mathcal{A}[\cap_{n=1}^\infty A_n\in\mathcal{A}]</math>
|-
| closed under complement<ref name="PTACC"/>
| CLOSED_complement
| If {{M|\mathcal{A} }} is closed under [[Complement|complement]] then <math>\forall A\in\mathcal{A}[A^c\in\mathcal{A}]</math>
|-
| {{M|\cup}}-closed<ref name="PTACC"/>
| CLOSED_cup
| If {{M|\mathcal{A} }} is ''{{M|\cup}}-closed'' then <math>\forall A,B\in\mathcal{A}[A\cup B\in\mathcal{A}]</math> - {{M|\mathcal{A} }} is closed under finite [[Union|union]]
|-
| {{Sigma|{{M|\cup}}-closed}}<ref name="PTACC"/>
| CLOSED_cup_sigma
| closed under [[Countably infinite|countably infinite]] union. <math>\forall (A_n)_{n=1}^\infty\subseteq\mathcal{A}[\cup_{n=1}^\infty A_n\in\mathcal{A}]</math>
|-
| {{M|\backslash}}-closed
| CLOSED_division
| See <code>CLOSED_backslash</code>
|}


==Notes==
<references group="Note"/>

==References==
<references/>

[[Category:Index]]