Theorems · Theorem · general topology
IsCompact.nonempty_iInter_of_sequence_nonempty_isCompact_isClosed
∀ {X : Type u} [inst : TopologicalSpace X] (t : ℕ → Set X),
(∀ (i : ℕ), t (i + 1) ⊆ t i) →
(∀ (i : ℕ), (t i).Nonempty) → IsCompact (t 0) → (∀ (i : ℕ), IsClosed (t i)) → (⋂ i, t i).NonemptyCantor's intersection theorem for sequences indexed by ℕ:
the intersection of a decreasing sequence of nonempty compact closed sets is nonempty.
- Defined in
- Mathlib.Topology.Compactness.Compact
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 79 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
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.Nonemptystatement and proof · cited by 2,627
- IsClosedstatement and proof · cited by 1,639
- IsCompactstatement and proof · cited by 1,282
- Set.iInterstatement · cited by 1,084
- Antitoneproof · cited by 563
- Directedproof · cited by 213
- IsCompact.of_isClosed_subsetproof · cited by 67
- antitone_nat_of_succ_leproof · cited by 21
- IsCompact.nonempty_iInter_of_directed_nonempty_isCompact_isClosedproof · cited by 11
- Antitone.directed_geproof · cited by 5
Cited by4
Results whose statement or proof uses this declaration.
- isCompactSystem_isCompact_isClosedproof · cited by 2
- Hindman.exists_idempotent_ultrafilter_le_FPproof · cited by 1
- Hindman.exists_idempotent_ultrafilter_le_FSproof · cited by 1