Theorems · Theorem · general topology
isCompact_pi_infinite
- 1000+ list: Tychonoff's theorem
∀ {ι : Type u_1} {X : ι → Type u_2} [inst : (i : ι) → TopologicalSpace (X i)] {s : (i : ι) → Set (X i)},
(∀ (i : ι), IsCompact (s i)) → IsCompact {x | ∀ (i : ι), x i ∈ s i}Tychonoff's theorem: product of compact sets is compact.
- Defined in
- Mathlib.Topology.Compactness.Compact
- Cited by
- 3 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.
Cites13
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.ofPredstatement and proof · cited by 6,101
- nhdsproof · cited by 5,554
- Filter.Tendstoproof · cited by 3,814
- IsCompactstatement · cited by 1,282
- Filter.mem_of_supersetproof · cited by 308
- Ultrafilterproof · cited by 193
- Ultrafilter.toFilterproof · cited by 172
- Function.evalproof · cited by 140
- Filter.mem_mapproof · cited by 63
- nhds_piproof · cited by 29
Cited by3
Results whose statement or proof uses this declaration.
- isCompact_univ_piproof · cited by 9
- ContinuousLinearMap.isCompact_closure_image_coe_of_boundedproof · cited by 1
- IsCompact.matrixproof · cited by 0