Theorems · Theorem · general topology
Convex.isConnected
∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] [inst_2 : TopologicalSpace E] [ContinuousAdd E]
[ContinuousSMul ℝ E] {s : Set E}, Convex ℝ s → s.Nonempty → IsConnected sA nonempty convex set is connected.
- Defined in
- Mathlib.Analysis.Convex.PathConnected
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 133 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
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
- Set.Nonemptystatement and proof · cited by 2,627
- ContinuousSMulstatement and proof · cited by 1,016
- ContinuousAddstatement and proof · cited by 777
- Convexstatement and proof · cited by 551
- IsConnectedstatement · cited by 116
- Convex.isPathConnectedproof · cited by 8
- IsPathConnected.isConnectedproof · cited by 7
Cited by3
Results whose statement or proof uses this declaration.
- Convex.isPreconnectedproof · cited by 14
- Complex.isConnected_of_upperHalfPlaneproof · cited by 2
- Complex.isConnected_of_lowerHalfPlaneproof · cited by 1