Theorems · Theorem · general topology
TopologicalSpace.IsTopologicalBasis.eq_generateFrom
∀ {α : Type u} [t : TopologicalSpace α] {s : Set (Set α)},
TopologicalSpace.IsTopologicalBasis s → t = TopologicalSpace.generateFrom sThe topology is generated by sets from s.
- Defined in
- Mathlib.Topology.Bases
- Cited by
- 10 results in Mathlib
- Foundations
- Depth 8 from the axioms · uses no axioms
- Assumes
- TopologicalSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites4
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
- TopologicalSpace.IsTopologicalBasisstatement and proof · cited by 126
- TopologicalSpace.generateFromstatement · cited by 62
Cited by10
Results whose statement or proof uses this declaration.
- TopologicalSpace.IsTopologicalBasis.isOpenproof · cited by 31
- TopologicalSpace.IsTopologicalBasis.mem_nhds_iffproof · cited by 13
- TopologicalSpace.IsTopologicalBasis.continuous_iffproof · cited by 4
- Topology.IsQuotientMap.secondCountableTopologyproof · cited by 1
- TopologicalSpace.eq_generateFrom_countableBasisproof · cited by 1
- TopologicalSpace.IsTopologicalBasis.secondCountableTopologyproof · cited by 1
- TopologicalSpace.Opens.IsBasis.le_iffproof · cited by 1
- TopologicalSpace.exists_countable_of_generateFromproof · cited by 1
- TopologicalSpace.IsTopologicalBasis.borel_eq_generateFromproof · cited by 1
- TopologicalSpace.Clopens.countable_iff_secondCountableproof · cited by 0