Types of set algebras

Relationship between types

This diagram shows the relations between types of system:

$$\begin{xy}\xymatrix{ & & \text{Dynkin system} \ar[d]^{\cap\text{-closed}} & \\ & {\sigma\text{-ring}} \ar[r]^{\Omega\in\mathcal{A}} & {\sigma\text{-algebra}} & \\ \text{ring} \ar[ur]^{\sigma\text{-}\cup} \ar[r]^{\Omega\in\mathcal{A}} & \text{algebra} \ar[ur]_{\sigma\text{-}\cup} & & \text{topology}\ar@{.>}[ul]_(.25){\text{Borel }\sigma\text{-algebra}} \\ \text{semiring} \ar[u]_{\cup\text{-closed}} & & & }\end{xy}$$

Diagram showing the relationships

Notes

Closed under
Type	$\sigma\in\mathcal{A}$	$\bigcap$	$\sigma$-$\bigcap$	$\bigcup$	$\sigma$-$\bigcup$	$-$	$C$
Semi-Ring
Ring
$\sigma$-Ring
Algebra
Dynkin system
$\sigma$-Algebra	#	#		X	X	#