Theorems · Theorem · general topology
isOpen_sigma_iff
∀ {ι : Type u_5} {σ : ι → Type u_7} [inst : (i : ι) → TopologicalSpace (σ i)] {s : Set (Sigma σ)},
IsOpen s ↔ ∀ (i : ι), IsOpen (Sigma.mk i ⁻¹' s)- Defined in
- Mathlib.Topology.Constructions
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 66 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.
Cites5
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
- IsOpenstatement and proof · cited by 2,400
- isOpen_iSup_iffproof · cited by 4
Cited by4
Results whose statement or proof uses this declaration.
- isOpenMap_sigmaMkproof · cited by 5
- IsCompactOpenCovered.iff_isCompactOpenCovered_sigmaMkproof · cited by 3
- UCompactlyGeneratedSpace.isOpenproof · cited by 1
- Metric.Sigma.isOpen_iffproof · cited by 0