Theorems · Theorem · general topology
isOpen_set_pi
∀ {ι : Type u_5} {A : ι → Type u_6} [T : (i : ι) → TopologicalSpace (A i)] {i : Set ι} {s : (a : ι) → Set (A a)},
i.Finite → (∀ a ∈ i, IsOpen (s a)) → IsOpen (i.pi s)- Defined in
- Mathlib.Topology.Constructions
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 81 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.
Cites9
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
- IsOpenstatement and proof · cited by 2,400
- Set.Finitestatement and proof · cited by 1,814
- Set.pistatement · cited by 405
- IsOpen.preimageproof · cited by 147
- continuous_applyproof · cited by 120
- Set.pi_defproof · cited by 14
- Set.Finite.isOpen_biInterproof · cited by 10
Cited by8
Results whose statement or proof uses this declaration.
- SymbolicDynamics.FullShift.isOpen_cylinderproof · cited by 2
- ZLattice.exists_forall_abs_repr_le_normproof · cited by 2
- RestrictedProduct.isOpen_forall_imp_mem_of_principalproof · cited by 2
- pi_generateFrom_eqproof · cited by 2
- PiNat.isOpen_cylinderproof · cited by 2
- Topology.IsOpenEmbedding.piMapproof · cited by 1
- IsOpen.matrixproof · cited by 0
- MeasureTheory.ProbabilityMeasure.continuous_piproof · cited by 0