Theorems · Theorem · general topology
isOpen_sUnion
∀ {X : Type u} [inst : TopologicalSpace X] {s : Set (Set X)}, (∀ t ∈ s, IsOpen t) → IsOpen (⋃₀ s)- Defined in
- Mathlib.Topology.Defs.Basic
- Cited by
- 15 results in Mathlib
- Foundations
- Depth 7 from the axioms · uses no axioms
- 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
- IsOpenstatement and proof · cited by 2,400
- Set.sUnionstatement · cited by 392
- TopologicalSpace.isOpen_sUnionproof · cited by 1
Cited by15
Results whose statement or proof uses this declaration.
- isOpen_interiorproof · cited by 130
- isOpen_iUnionproof · cited by 88
- isOpen_emptyproof · cited by 23
- TopologicalSpace.le_generateFrom_iff_subset_isOpenproof · cited by 5
- TopologicalSpace.vietoris.isTopologicalBasisproof · cited by 2
- TopologicalSpace.IsTopologicalBasis.compactsproof · cited by 2
- MvPolynomial.isOpenMap_comap_Cproof · cited by 1
- TopologicalSpace.IsTopologicalBasis.nonemptyCompactsproof · cited by 1
- TopologicalSpace.IsTopologicalBasis.open_iff_eq_sUnionproof · cited by 1
- TopologicalSpace.IsTopologicalBasis.vietorisproof · cited by 1
- IsClosed.exists_minimal_nonempty_closed_subsetproof · cited by 1
- ContinuousMap.compactOpen_eq_generateFromproof · cited by 1