Theorems · Theorem · general topology
IsOpen.isClosed_compl
∀ {X : Type u} [inst : TopologicalSpace X] {s : Set X}, IsOpen s → IsClosed sᶜAlias of the reverse direction of isClosed_compl_iff.
- Defined in
- Mathlib.Topology.Basic
- Cited by
- 50 results in Mathlib
- Foundations
- Depth 58 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.
Cites6
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
- Compl.complstatement · cited by 2,925
- IsOpenstatement · cited by 2,400
- IsClosedstatement · cited by 1,639
- isClosed_compl_iffproof · cited by 35
Cited by50
Results whose statement or proof uses this declaration.
- IsClopen.complproof · cited by 12
- IsOpen.inter_closureproof · cited by 8
- isPreconnected_closed_iffproof · cited by 6
- Disjoint.closure_leftproof · cited by 6
- AlgebraicGeometry.exists_map_eq_topproof · cited by 5
- exists_continuous_one_zero_of_isCompactproof · cited by 5
- MeasureTheory.Measure.WeaklyRegular.innerRegular_measurableproof · cited by 4
- spectrum.isClosedproof · cited by 4
- IsCompact.exists_cthickening_subset_openproof · cited by 4
- isLindelof_of_countable_subfamily_closedproof · cited by 2
- connectedComponent_eq_iInter_isClopenproof · cited by 2
- MeasureTheory.exists_continuous_eLpNorm_sub_le_of_closedproof · cited by 2