Theorems · Theorem · general topology
jacobsonSpace_iff_locallyClosed
∀ {X : Type u_1} [inst : TopologicalSpace X],
JacobsonSpace X ↔ ∀ (Z : Set X), Z.Nonempty → IsLocallyClosed Z → (Z ∩ closedPoints X).Nonempty- Defined in
- Mathlib.Topology.JacobsonSpace
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 77 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.
Cites31
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
- LE.le.transproof · cited by 3,151
- Compl.complproof · cited by 2,925
- Set.Nonemptystatement and proof · cited by 2,627
- IsClosedproof · cited by 1,639
- closureproof · cited by 1,254
- Set.inter_subset_leftproof · cited by 360
- Set.inter_subset_rightproof · cited by 329
- subset_closureproof · cited by 309
- Set.inter_commproof · cited by 291
- isClosed_closureproof · cited by 195
Cited by4
Results whose statement or proof uses this declaration.
- nonempty_inter_closedPointsproof · cited by 5
- JacobsonSpace.of_isOpenEmbeddingproof · cited by 3
- TopologicalSpace.IsOpenCover.jacobsonSpace_iffproof · cited by 1
- JacobsonSpace.of_isClosedEmbeddingproof · cited by 1