Theorems · Theorem · general topology
isLUB_iff_of_subset_of_subset_closure
∀ {α : Type u_3} [inst : TopologicalSpace α] [inst_1 : Preorder α] [ClosedIicTopology α] {s t : Set α},
s ⊆ t → t ⊆ closure s → ∀ {x : α}, IsLUB s x ↔ IsLUB t x- Defined in
- Mathlib.Topology.Order.IsLUB
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 64 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- Preorderstatement and proof · cited by 7,952
- closurestatement and proof · cited by 1,254
- LE.le.antisymmproof · cited by 507
- IsLUBstatement · cited by 280
- ClosedIicTopologystatement and proof · cited by 115
- upperBounds_mono_setproof · cited by 13
- isLUB_congrproof · cited by 4
- upperBounds_closureproof · cited by 2
Cited by2
Results whose statement or proof uses this declaration.
- Dense.isLUB_inter_iffproof · cited by 2
- isGLB_iff_of_subset_of_subset_closureproof · cited by 0