Types of set algebras

Relationship between types

This diagram shows the relations between types of system:

[math]\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}[/math]

Diagram showing the relationships

Notes

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