Theorems · Theorem · general topology
isClosed_sigma_iff
∀ {ι : Type u_5} {σ : ι → Type u_7} [inst : (i : ι) → TopologicalSpace (σ i)] {s : Set (Sigma σ)},
IsClosed s ↔ ∀ (i : ι), IsClosed (Sigma.mk i ⁻¹' s)- Defined in
- Mathlib.Topology.Constructions
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 68 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.
Cites6
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.preimagestatement and proof · cited by 4,946
- Compl.complproof · cited by 2,925
- IsOpenproof · cited by 2,400
- IsClosedstatement · cited by 1,639
Cited by2
Results whose statement or proof uses this declaration.
- UCompactlyGeneratedSpace.isClosedproof · cited by 2
- isClosedMap_sigmaMkproof · cited by 2