Theorems · Theorem · convex and discrete geometry
Convex.nontrivial_iff_nonempty_interior
∀ {𝕜 : Type u_4} [inst : Field 𝕜] [inst_1 : LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [inst_3 : TopologicalSpace 𝕜]
[OrderTopology 𝕜] {s : Set 𝕜}, Convex 𝕜 s → (s.Nontrivial ↔ (interior s).Nonempty)- Defined in
- Mathlib.Analysis.Convex.Topology
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 88 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites21
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
- LinearOrderstatement and proof · cited by 8,572
- Fieldstatement and proof · cited by 7,404
- Set.Nonemptystatement and proof · cited by 2,627
- IsStrictOrderedRingstatement and proof · cited by 2,490
- OrderTopologystatement and proof · cited by 1,355
- interiorstatement and proof · cited by 714
- Convexstatement and proof · cited by 551
- interior_subsetproof · cited by 171
- Set.Nontrivialstatement and proof · cited by 145
- segmentproof · cited by 120
Cited by2
Results whose statement or proof uses this declaration.
- Convex.Ioo_subset_of_mem_closureproof · cited by 3
- taylor_isLittleOproof · cited by 2