Theorems · Theorem · general topology
isClosed_compl_iff
∀ {X : Type u} [inst : TopologicalSpace X] {s : Set X}, IsClosed sᶜ ↔ IsOpen s- Defined in
- Mathlib.Topology.Basic
- Cited by
- 35 results in Mathlib
- Foundations
- Depth 57 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
- Compl.complstatement · cited by 2,925
- IsOpenstatement and proof · cited by 2,400
- IsClosedstatement · cited by 1,639
- compl_complproof · cited by 229
- isOpen_compl_iffproof · cited by 63
Cited by35
Results whose statement or proof uses this declaration.
- IsOpen.isClosed_complproof · cited by 50
- mem_closure_iffproof · cited by 15
- IsCompact.diffproof · cited by 8
- IsClosed.sdiffproof · cited by 7
- normal_exists_closure_subsetproof · cited by 5
- measurable_of_isClosedproof · cited by 4
- MeasureTheory.ProbabilityMeasure.le_liminf_measure_open_of_tendstoproof · cited by 3
- borel_eq_generateFrom_isClosedproof · cited by 3
- isClopen_inter_of_disjoint_cover_clopenproof · cited by 3
- nhdsNE_of_nhdsNE_sdiff_finiteproof · cited by 3
- mem_codiscrete'proof · cited by 2
- BumpCovering.exists_isSubordinate_of_locallyFinite_of_propproof · cited by 2