Theorems · Theorem · general topology
Convex.isPathConnected
∀ {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 → IsPathConnected sA nonempty convex set is path connected.
- Defined in
- Mathlib.Analysis.Convex.PathConnected
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 124 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
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
- IsPathConnectedstatement and proof · cited by 65
- StarConvex.isPathConnectedproof · cited by 1
Cited by8
Results whose statement or proof uses this declaration.
- Convex.isConnectedproof · cited by 3
- Submodule.isPathConnectedproof · cited by 1
- Metric.isPathConnected_ballproof · cited by 1
- Metric.isPathConnected_eballproof · cited by 1
- Convex.locallyPathConnectedSpaceproof · cited by 1
- Metric.isPathConnected_closedBallproof · cited by 0
- isPathConnected_stdSimplexproof · cited by 0
- IsTopologicalAddGroup.pathConnectedSpaceproof · cited by 0