Theorems · Theorem · general topology
IsClosed.isOpen_compl
∀ {X : Type u} {inst : TopologicalSpace X} {s : Set X} [self : IsClosed s], IsOpen sᶜThe complement of a closed set is an open set.
- Defined in
- Mathlib.Topology.Defs.Basic
- Cited by
- 126 results in Mathlib
- Foundations
- Depth 5 from the axioms, rests on 16 definitions · uses no axioms
- Assumes
- IsClosed
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
- Compl.complstatement · cited by 2,925
- IsOpenstatement · cited by 2,400
- IsClosedstatement and proof · cited by 1,639
Cited by126
Results whose statement or proof uses this declaration.
- IsClosed.measurableSetproof · cited by 65
- isOpen_compl_iffproof · cited by 63
- isOpen_Ioiproof · cited by 43
- isOpen_Iioproof · cited by 38
- isOpen_compl_singletonproof · cited by 27
- isOpen_ltproof · cited by 23
- mem_closure_iffproof · cited by 15
- IsClopen.complproof · cited by 12
- exists_continuous_zero_one_of_isClosedproof · cited by 10
- IsClosed.compl_mem_nhdsproof · cited by 9
- IsOpen.sdiffproof · cited by 8
- t1Space_TFAEproof · cited by 8