Theorems · Theorem · general topology
TopologicalSpace.vietoris.isPreconnected_biUnion
∀ {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] {s : Set α} {f : α → Set β},
IsPreconnected s → ContinuousOn f s → (∃ x ∈ s, IsPreconnected (f x)) → IsPreconnected (⋃ x ∈ s, f x)- Defined in
- Mathlib.Topology.Sets.VietorisTopology
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 78 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
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.iUnionstatement · cited by 2,483
- ContinuousOnstatement and proof · cited by 1,411
- IsPreconnectedstatement and proof · cited by 205
- TopologicalSpace.vietorisstatement · cited by 34
- Set.sUnion_imageproof · cited by 28
- IsPreconnected.imageproof · cited by 24
- TopologicalSpace.vietoris.isPreconnected_sUnionproof · cited by 1
Cited by1
Results whose statement or proof uses this declaration.
- TopologicalSpace.NonemptyCompacts.locallyConnectedSpace_iffproof · cited by 1