Theorems · Theorem · convex and discrete geometry
QuasiconcaveOn.isPreconnected_preimage_subtype
∀ {E : Type u_4} [inst : AddCommGroup E] [inst_1 : Module ℝ E] [inst_2 : TopologicalSpace E] [IsTopologicalAddGroup E]
[ContinuousSMul ℝ E] {β : Type u_5} [inst_5 : Preorder β] {f : E → β} {s : Set E} {t : β},
QuasiconcaveOn ℝ s f → IsPreconnected (Subtype.val ⁻¹' f ⁻¹' Set.Ici t)If f is quasiconcave, then its over-levels are connected.
- Defined in
- Mathlib.Analysis.Convex.Quasiconvex
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 135 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites19
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
- Realstatement and proof · cited by 25,697
- TopologicalSpacestatement and proof · cited by 24,529
- Modulestatement and proof · cited by 20,661
- AddCommGroupstatement and proof · cited by 12,871
- Preorderstatement and proof · cited by 7,952
- Set.Elemstatement · cited by 7,166
- Set.preimagestatement and proof · cited by 4,946
- IsTopologicalAddGroupstatement and proof · cited by 1,394
- Set.Icistatement and proof · cited by 1,070
- ContinuousSMulstatement and proof · cited by 1,016
- Set.inter_commproof · cited by 291
Cited by1
Results whose statement or proof uses this declaration.
- QuasiconvexOn.isPreconnected_preimage_subtypeproof · cited by 1