Theorems · Theorem · general topology
isPathConnected_compl_singleton_of_one_lt_rank
∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] [inst_2 : TopologicalSpace E] [ContinuousAdd E]
[ContinuousSMul ℝ E], 1 < Module.rank ℝ E → ∀ (x : E), IsPathConnected {x}ᶜIn a real vector space of dimension > 1, the complement of any singleton is path-connected.
- Defined in
- Mathlib.Analysis.Normed.Module.Connected
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 171 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · 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
- Compl.complstatement · cited by 2,925
- Cardinalstatement · cited by 2,598
- ContinuousSMulstatement and proof · cited by 1,016
- ContinuousAddstatement and proof · cited by 777
- Module.rankstatement and proof · cited by 496
- IsPathConnectedstatement · cited by 65
- Set.countable_singletonproof · cited by 13
Cited by3
Results whose statement or proof uses this declaration.
- isConnected_compl_singleton_of_one_lt_rankproof · cited by 2
- isPathConnected_compl_of_one_lt_codimproof · cited by 1
- isPathConnected_sphereproof · cited by 1