Theorems · Theorem · general topology
IsOpen.powerset_vietoris
∀ {α : Type u_1} [inst : TopologicalSpace α] {U : Set α}, IsOpen U → IsOpen (𝒫 U)When Set is equipped with the Vietoris topology, the powerset of an open set is open.
- Defined in
- Mathlib.Topology.Sets.VietorisTopology
- Cited by
- 11 results in Mathlib
- Foundations
- Depth 9 from the axioms · uses no axioms
- Assumes
- TopologicalSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
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
- IsOpenstatement and proof · cited by 2,400
- Set.powersetstatement · cited by 67
- TopologicalSpace.vietorisstatement · cited by 34
- TopologicalSpace.isOpen_generateFrom_of_memproof · cited by 11
Cited by11
Results whose statement or proof uses this declaration.
- TopologicalSpace.Compacts.isOpen_subsets_of_isOpenproof · cited by 7
- TopologicalSpace.NonemptyCompacts.isOpen_subsets_of_isOpenproof · cited by 4
- TopologicalSpace.vietoris.isClopen_singleton_emptyproof · cited by 2
- TopologicalSpace.vietoris.isClosed_inter_nonempty_of_isClosedproof · cited by 2
- TopologicalSpace.vietoris.isTopologicalBasisproof · cited by 2
- Continuous.image_vietorisproof · cited by 1
- TopologicalSpace.IsTopologicalBasis.vietorisproof · cited by 1
- TotallyBounded.nhds_vietoris_le_nhds_hausdorffproof · cited by 1
- TopologicalSpace.vietoris.isPreconnected_sUnionproof · cited by 1
- TopologicalSpace.vietoris.continuous_iffproof · cited by 0
- TopologicalSpace.vietoris.continuous_unionproof · cited by 0