Theorems · Theorem · general topology
closure_minimal
∀ {X : Type u} [inst : TopologicalSpace X] {s t : Set X}, s ⊆ t → IsClosed t → closure s ⊆ t- Defined in
- Mathlib.Topology.Closure
- Cited by
- 94 results in Mathlib
- Foundations
- Depth 61 from the axioms, rests on 681 definitions · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
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
- IsClosedstatement and proof · cited by 1,639
- closurestatement · cited by 1,254
- Set.sInter_subset_of_memproof · cited by 28
Cited by94
Results whose statement or proof uses this declaration.
- IsClosed.closure_eqproof · cited by 139
- closure_monoproof · cited by 133
- IsClosed.closure_subset_iffproof · cited by 59
- closure_Iooproof · cited by 20
- mem_closure_iffproof · cited by 15
- IsClosed.closure_subsetproof · cited by 11
- Metric.closure_ball_subset_closedBallproof · cited by 10
- IsClosedMap.closure_image_subsetproof · cited by 7
- Dense.inter_of_isOpen_leftproof · cited by 7
- Set.EqOn.closureproof · cited by 6
- Disjoint.closure_leftproof · cited by 6
- StarSubalgebra.topologicalClosure_minimalproof · cited by 6