Difference between revisions of "Index of properties"

Revision as of 20:40, 15 June 2015

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

Adjective	Specific case	Index	Description
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
	$\backslash$ -closed	CLOSED_backslash	To say $\mathcal{A}$ is $\backslash$ -closed uses $\backslash$ to denote set subtraction^{[Note 1]}, this means $\forall A,B\in\mathcal{A}[A-B\in\mathcal{A}]$
	$\cap$ -closed	CLOSED_cap	If $\mathcal{A}$ is $\cap$ -closed then - $\mathcal{A}$ is closed under finite intersection
	$\sigma$ - $\cap$ -closed	CLOSED_cap_sigma	closed under countably infinite intersection. $\forall (A_n)_{n=1}^\infty\subseteq\mathcal{A}[\cap_{n=1}^\infty A_n\in\mathcal{A}]$
	closed under complement	CLOSED_complement	If $\mathcal{A}$ is closed under complement then $\forall A\in\mathcal{A}[A^c\in\mathcal{A}]$
	$\cup$ -closed	CLOSED_cup	If $\mathcal{A}$ is $\cup$ -closed then - $\mathcal{A}$ is closed under finite union
	$\sigma$ - $\cup$ -closed	CLOSED_cup_sigma	closed under countably infinite union. $\forall (A_n)_{n=1}^\infty\subseteq\mathcal{A}[\cup_{n=1}^\infty A_n\in\mathcal{A}]$
	$\backslash$ -closed	CLOSED_division	See `CLOSED_backslash`