Theorems · Theorem · general topology
IsClosed.exists_nat_bool_injection_of_not_countable
∀ {α : Type u_1} [inst : TopologicalSpace α] [PolishSpace α] {C : Set α},
IsClosed C → ¬C.Countable → ∃ f, Set.range f ⊆ C ∧ Continuous f ∧ Function.Injective fAny closed uncountable subset of a Polish space admits a continuous injection
from the Cantor space ℕ → Bool.
- Defined in
- Mathlib.Topology.MetricSpace.Perfect
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 156 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpacePolishSpace
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Cites14
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
- TopologicalSpacestatement and proof · cited by 24,529
- Set.rangestatement and proof · cited by 4,705
- LE.le.transproof · cited by 3,151
- Set.Nonemptyproof · cited by 2,627
- Continuousstatement and proof · cited by 2,592
- IsClosedstatement and proof · cited by 1,639
- Set.Countablestatement and proof · cited by 545
- PolishSpacestatement and proof · cited by 57
- Perfectproof · cited by 17
- TopologicalSpace.UpgradedIsCompletelyMetrizableSpaceproof · cited by 7
- TopologicalSpace.upgradeIsCompletelyMetrizableproof · cited by 7
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