Hereditary system of sets

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Needs linking to where it is used, notes on a sort of "power-set" like construct.

Note: see hereditary for different uses of the word, this page refers to hereditary as used in Measure Theory and Hereditary (measure theory) redirects here

Definition

A collection of sets, $\mathcal{H}$ is said to be hereditary if:

• $\forall A\in\mathcal{H}\forall B\in\mathcal{P}(A)[B\in\mathcal{H}]$, in words:
  • for all sets $A\in\mathcal{H}$ all subsets of $A$ must be in $\mathcal{H}$

See also

• Hereditary $\sigma$-ring - the motivation for this definition.