Theorems · Theorem · general topology
completelyNormalSpace_iff_forall_normalSpace
∀ {X : Type u_1} [inst : TopologicalSpace X], CompletelyNormalSpace X ↔ ∀ (s : Set X), NormalSpace ↑sA space is completely normal iff it is hereditarily normal.
- Defined in
- Mathlib.Topology.Separation.Regular
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 76 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
- Set.Elemstatement and proof · cited by 7,166
- IsOpenproof · cited by 2,400
- NormalSpacestatement and proof · cited by 84
- CompletelyNormalSpacestatement and proof · cited by 9
- completelyNormalSpace_iff_forall_isOpen_normalSpaceproof · cited by 3
Cited by1
Results whose statement or proof uses this declaration.
- CompletelyNormalSpace.of_forall_normalSpaceproof · cited by 0