Theorems · Theorem · general topology
IsPreconnected.subset_left_of_subset_union
∀ {α : Type u} [inst : TopologicalSpace α] {s u v : Set α},
IsOpen u → IsOpen v → Disjoint u v → s ⊆ u ∪ v → (s ∩ u).Nonempty → IsPreconnected s → s ⊆ u- Defined in
- Mathlib.Topology.Connected.Basic
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 60 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.
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.Nonemptystatement and proof · cited by 2,627
- IsOpenstatement and proof · cited by 2,400
- Disjointstatement and proof · cited by 2,201
- IsPreconnectedstatement and proof · cited by 205
- Set.not_disjoint_iff_nonempty_interproof · cited by 29
- Set.disjoint_iffproof · cited by 27
- Disjoint.subset_left_of_subset_unionproof · cited by 2
Cited by5
Results whose statement or proof uses this declaration.
- IsPreconnected.subset_isClopenproof · cited by 4
- IsPreconnected.subset_of_closure_inter_subsetproof · cited by 3
- Complex.norm_eqOn_of_isPreconnected_of_isMaxOnproof · cited by 3
- MDifferentiableOn.norm_eqOn_of_isPreconnected_of_isMaxOnproof · cited by 2
- IsPreconnected.subset_right_of_subset_unionproof · cited by 0