Theorems · Theorem · order theory
ClosureOperator.le_closure
∀ {α : Type u_1} [inst : Preorder α] (c : ClosureOperator α) (x : α), x ≤ c xEvery element is less than its closure. This property is sometimes referred to as extensivity or inflationarity.
- Defined in
- Mathlib.Order.Closure
- Cited by
- 19 results in Mathlib
- Foundations
- Depth 14 from the axioms · uses propext, Quot.sound
- Assumes
- Preorder
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites4
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- Preorderstatement and proof · cited by 7,952
- ClosureOperatorstatement and proof · cited by 371
- ClosureOperator.le_closure'proof · cited by 1
Cited by19
Results whose statement or proof uses this declaration.
- subset_convexHullproof · cited by 36
- subset_supClosureproof · cited by 11
- subset_absConvexHullproof · cited by 7
- subset_countableInfClosureproof · cited by 7
- subset_countableSupClosureproof · cited by 7
- subset_latticeClosureproof · cited by 6
- subset_infClosureproof · cited by 6
- ClosureOperator.closure_sup_closure_leftproof · cited by 5
- Nucleus.le_applyproof · cited by 5
- ClosureOperator.le_closure_iffproof · cited by 5
- ClosureOperator.closure_topproof · cited by 4
- ClosureOperator.isClosed_iff_closure_leproof · cited by 3