Theorems · Theorem · general topology
Perfect.exists_nat_bool_injection
∀ {α : Type u_1} [inst : MetricSpace α] {C : Set α},
Perfect C → C.Nonempty → ∀ [CompleteSpace α], ∃ f, Set.range f ⊆ C ∧ Continuous f ∧ Function.Injective fAny nonempty perfect set in a complete metric space admits a continuous injection
from the Cantor space, ℕ → Bool.
- Defined in
- Mathlib.Topology.MetricSpace.Perfect
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 155 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- MetricSpaceCompleteSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites39
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
- ENNRealproof · cited by 9,879
- nhdsproof · cited by 5,554
- Set.rangestatement and proof · cited by 4,705
- Filter.Tendstoproof · cited by 3,814
- Filter.Eventuallyproof · cited by 3,134
- Set.Nonemptystatement and proof · cited by 2,627
- Continuousstatement · cited by 2,592
- CompleteSpacestatement and proof · cited by 2,532
- Filter.atTopproof · cited by 2,405
- Disjointproof · cited by 2,201
- MetricSpacestatement and proof · cited by 1,684
Cited by2
Results whose statement or proof uses this declaration.
- continuum_le_cardinal_of_nontriviallyNormedFieldproof · cited by 1
- IsClosed.exists_nat_bool_injection_of_not_countableproof · cited by 0