Theorems · Theorem · general topology
isConnected_sphere
∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E],
1 < Module.rank ℝ E → ∀ (x : E) {r : ℝ}, 0 ≤ r → IsConnected (Metric.sphere x r)In a real vector space of dimension > 1, any sphere of nonnegative radius is connected.
- Defined in
- Mathlib.Analysis.Normed.Module.Connected
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 173 from the axioms · uses propext, Classical.choice, Quot.sound
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.
- Realstatement and proof · cited by 25,697
- NormedAddCommGroupstatement and proof · cited by 15,752
- NormedSpacestatement and proof · cited by 12,499
- Cardinalstatement · cited by 2,598
- Module.rankstatement and proof · cited by 496
- Metric.spherestatement · cited by 371
- IsConnectedstatement · cited by 116
- IsPathConnected.isConnectedproof · cited by 7
- isPathConnected_sphereproof · cited by 1
Cited by2
Results whose statement or proof uses this declaration.
- MeromorphicOn.circleIntegrable_log_normproof · cited by 11
- isPreconnected_sphereproof · cited by 0