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:
    • [math]\mathcal{H}(S):=\bigcap_{\text{All hereditary systems of sets, }\mathcal{H}\text{, where }S\subseteq\mathcal{H} }\mathcal{H}[/math]
    This is extremely similar to sigma-ring generated by and many other generators involving systems of sets - there is certainly something to abstract here.
    Grade: A
    This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.
    Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).
    The message provided is:
    Prove that the intersection of an arbitrary family of hereditary systems of sets (among other systems of course!) is itself a hereditary system of sets
  2. There are many ways to write this and this may not be the best. The "defining property" (Note to self: explore notion between FOL sentences and sets) is:
    • [ilmath]\left[A\in\mathcal{H}(S)\right]\iff[\exists B\in S(A\in\mathcal{P}(B))][/ilmath]

References

  1. Measure Theory - Paul R. Halmos