Theorems · Theorem · general topology
TopologicalSpace.vietoris.isClosed_inter_nonempty_of_isClosed
∀ {α : Type u_1} [inst : TopologicalSpace α] {F : Set α}, IsClosed F → IsClosed {s | (s ∩ F).Nonempty}- Defined in
- Mathlib.Topology.Sets.VietorisTopology
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 59 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.
Cites10
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.ofPredstatement and proof · cited by 6,101
- Set.Nonemptystatement and proof · cited by 2,627
- IsClosedstatement and proof · cited by 1,639
- compl_complproof · cited by 229
- IsClosed.isOpen_complproof · cited by 126
- IsOpen.isClosed_complproof · cited by 50
- TopologicalSpace.vietorisstatement · cited by 34
- IsOpen.powerset_vietorisproof · cited by 11
Cited by2
Results whose statement or proof uses this declaration.
- TopologicalSpace.vietoris.specializes_iffproof · cited by 0
- TopologicalSpace.NonemptyCompacts.isClosed_inter_nonempty_of_isClosedproof · cited by 0