# Documentation

Mathlib.Data.Set.Constructions

# Constructions involving sets of sets. #

## Finite Intersections #

We define a structure FiniteInter which asserts that a set S of subsets of α is closed under finite intersections.

We define finiteInterClosure which, given a set S of subsets of α, is the smallest set of subsets of α which is closed under finite intersections.

finiteInterClosure S is endowed with a term of type FiniteInter using finiteInterClosure_finiteInter.

structure FiniteInter {α : Type u_1} (S : Set (Set α)) :
• univ_mem states that Set.univ is in S.

univ_mem : Set.univ S
• inter_mem states that any two intersections of sets in S is also in S.

inter_mem : ∀ ⦃s : Set α⦄, s S∀ ⦃t : Set α⦄, t Ss t S

A structure encapsulating the fact that a set of sets is closed under finite intersection.

Instances For
inductive FiniteInter.finiteInterClosure {α : Type u_1} (S : Set (Set α)) :
Set (Set α)

The smallest set of sets containing S which is closed under finite intersections.

Instances For
theorem FiniteInter.finiteInter_mem {α : Type u_1} {S : Set (Set α)} (cond : ) (F : Finset (Set α)) :
F S⋂₀ F S
theorem FiniteInter.finiteInterClosure_insert {α : Type u_1} {S : Set (Set α)} {A : Set α} (cond : ) (P : Set α) (H : ) :
P S Q, Q S P = A Q