Theorems · Theorem · general topology
TopologicalSpace.IsTopologicalBasis.isOpen_iff
∀ {α : Type u} [t : TopologicalSpace α] {s : Set α} {b : Set (Set α)},
TopologicalSpace.IsTopologicalBasis b → (IsOpen s ↔ ∀ a ∈ s, ∃ t ∈ b, a ∈ t ∧ t ⊆ s)- Defined in
- Mathlib.Topology.Bases
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 68 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.
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 · cited by 2,400
- TopologicalSpace.IsTopologicalBasisstatement and proof · cited by 126
- TopologicalSpace.IsTopologicalBasis.mem_nhds_iffproof · cited by 13
Cited by9
Results whose statement or proof uses this declaration.
- TopologicalSpace.IsTopologicalBasis.of_isOpen_of_subsetproof · cited by 4
- TopologicalSpace.IsTopologicalBasis.sdiff_emptyproof · cited by 3
- TopologicalSpace.IsTopologicalBasis.exists_mem_of_neproof · cited by 2
- totallySeparatedSpace_of_t0_of_basis_clopenproof · cited by 2
- PrimeSpectrum.toPiLocalization_surjective_of_discreteTopologyproof · cited by 2
- TopologicalSpace.IsTopologicalBasis.exists_countableproof · cited by 1
- Topology.IsOpenEmbedding.prespectralSpaceproof · cited by 1
- PrespectralSpace.exists_isClosed_of_not_isPreirreducibleproof · cited by 0
- PiNat.isOpen_iff_distproof · cited by 0