Theorems · Theorem · general topology
TopologicalSpace.IsTopologicalBasis.isOpen
∀ {α : Type u} [t : TopologicalSpace α] {s : Set α} {b : Set (Set α)},
TopologicalSpace.IsTopologicalBasis b → s ∈ b → IsOpen s- Defined in
- Mathlib.Topology.Bases
- Cited by
- 31 results in Mathlib
- Foundations
- Depth 9 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
- TopologicalSpace.IsTopologicalBasisstatement and proof · cited by 126
- TopologicalSpace.IsTopologicalBasis.eq_generateFromproof · cited by 10
Cited by31
Results whose statement or proof uses this declaration.
- AlgebraicGeometry.Scheme.isBasis_affineOpensproof · cited by 37
- ultrafilter_isOpen_basicproof · cited by 5
- TopologicalSpace.isOpen_of_mem_countableBasisproof · cited by 4
- TopologicalSpace.IsTopologicalBasis.isOpenMap_iffproof · cited by 4
- isCompact_open_iff_eq_finite_iUnion_of_isTopologicalBasisproof · cited by 4
- TopologicalSpace.IsTopologicalBasis.sdiff_emptyproof · cited by 3
- PrespectralSpace.of_isTopologicalBasisproof · cited by 3
- eq_finite_iUnion_of_isTopologicalBasis_of_isCompact_openproof · cited by 2
- Measurable.exists_continuousproof · cited by 2
- TopologicalSpace.IsTopologicalBasis.compactsproof · cited by 2
- TopologicalSpace.isTopologicalBasis_of_coverproof · cited by 2
- TopologicalSpace.IsTopologicalBasis.isQuotientMapproof · cited by 2