Theorems · Theorem · general topology
IsClosed.closure_subset_iff
∀ {X : Type u} [inst : TopologicalSpace X] {s t : Set X}, IsClosed t → (closure s ⊆ t ↔ s ⊆ t)- Defined in
- Mathlib.Topology.Closure
- Cited by
- 59 results in Mathlib
- Foundations
- Depth 62 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
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
- subset_closureproof · cited by 309
- Set.Subset.transproof · cited by 218
- closure_minimalproof · cited by 94
Cited by59
Results whose statement or proof uses this declaration.
- PrimeSpectrum.zeroLocus_vanishingIdeal_eq_closureproof · cited by 11
- specializes_TFAEproof · cited by 6
- AlgebraicGeometry.Scheme.Hom.support_kerproof · cited by 6
- HasCompactSupport.tsupport_extend_zero_subsetproof · cited by 5
- HasCompactMulSupport.mulTSupport_extend_one_subsetproof · cited by 4
- Topology.RelCWComplex.Subcomplex.closedCell_subset_of_memproof · cited by 4
- jacobsonSpace_iff_locallyClosedproof · cited by 4
- IsClosed.closure_interior_subsetproof · cited by 3
- Topology.RelCWComplex.closure_openCell_eq_closedCellproof · cited by 3
- Metric.hausdorffEDist_zero_iff_closure_eq_closureproof · cited by 3
- exists_contDiff_tsupport_subsetproof · cited by 3
- ContinuousLinearMap.is_weak_closed_closedBallproof · cited by 3