# Hereditary system generated by

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Demote grade once content is put in place, I really want to pull the "smallest" note out and give it a name (generated system of sets?) as this occurs a lot, and can be applied any time the type of system in question is closed under arbitrary intersection, that is the intersection of an arbitrary family of a type of system is a system of that type in and of itself.
Warning:This page is little more than notes at the moment, however everything stated here is verified and correct

## Definition

The hereditary system generated by a collection of sets, [ilmath]S[/ilmath], which we denote: [ilmath]\mathcal{H}(S)[/ilmath] is the smallest[Note 1] hereditary system containing [ilmath]S[/ilmath][1].

• Claim 1: [ilmath]\mathcal{H}(S)=\{V\in\mathcal{P}(T)\ \vert\ T\in S\}[/ilmath][Note 2] where [ilmath]\mathcal{P}(A)[/ilmath] denotes the power set of [ilmath]A[/ilmath].

## Proof of claims

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Pretty routine proof by contradiction. I've done it on some paper somewhere

## Notes

1. We do not mean smallest in the sense of cardinality arguments, we also do not mean smallest in the sense of [ilmath]\subset[/ilmath] relation (as given any two hereditary systems containing [ilmath]S[/ilmath] we cannot be sure that either one is a subset (proper or not) of the other! Instead we mean smallest in the following sense:
• $\mathcal{H}(S):=\bigcap_{\text{All hereditary systems of sets, }\mathcal{H}\text{, where }S\subseteq\mathcal{H} }\mathcal{H}$
This is extremely similar to sigma-ring generated by and many other generators involving systems of sets - there is certainly something to abstract here.