Theorems · Theorem · general topology
TopologicalSpace.IsTopologicalBasis.sdiff_empty
∀ {α : Type u} [t : TopologicalSpace α] {s : Set (Set α)},
TopologicalSpace.IsTopologicalBasis s → TopologicalSpace.IsTopologicalBasis (s \ {∅})- Defined in
- Mathlib.Topology.Bases
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 70 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.
Cites8
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
- IsOpenproof · cited by 2,400
- TopologicalSpace.IsTopologicalBasisstatement and proof · cited by 126
- Set.notMem_emptyproof · cited by 35
- TopologicalSpace.IsTopologicalBasis.isOpenproof · cited by 31
- TopologicalSpace.isTopologicalBasis_of_isOpen_of_nhdsproof · cited by 22
- TopologicalSpace.IsTopologicalBasis.isOpen_iffproof · cited by 9
Cited by3
Results whose statement or proof uses this declaration.
- TopologicalSpace.exists_countable_basisproof · cited by 9
- TopologicalSpace.IsTopologicalBasis.diff_emptyproof · cited by 0
- TopologicalSpace.isTopologicalBasis_singleton_emptyproof · cited by 0