Theorems · Theorem · general topology
Pi.isCompact_iff
∀ {ι : Type u_4} {X : ι → Type u_5} [inst : (i : ι) → TopologicalSpace (X i)] [∀ (i : ι), T2Space (X i)]
{s : Set ((i : ι) → X i)}, IsCompact s ↔ IsClosed s ∧ ∀ (i : ι), IsCompact (Function.eval i '' s)- Defined in
- Mathlib.Topology.Separation.Hausdorff
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 91 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpaceT2Space
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
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.imagestatement and proof · cited by 5,609
- Set.univproof · cited by 3,945
- IsClosedstatement and proof · cited by 1,639
- T2Spacestatement and proof · cited by 1,351
- IsCompactstatement and proof · cited by 1,282
- Function.evalstatement and proof · cited by 140
- continuous_applyproof · cited by 120
- IsCompact.imageproof · cited by 105
- IsCompact.isClosedproof · cited by 77
- IsCompact.of_isClosed_subsetproof · cited by 67
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