Theorems · Theorem · general topology
discreteTopology_iff_isOpen_singleton
∀ {α : Type u_1} [inst : TopologicalSpace α], DiscreteTopology α ↔ ∀ (a : α), IsOpen {a}- Defined in
- Mathlib.Topology.Order
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 66 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.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- IsOpenstatement and proof · cited by 2,400
- DiscreteTopologystatement and proof · cited by 373
- isOpen_discreteproof · cited by 36
- eq_bot_of_singletons_openproof · cited by 4
Cited by8
Results whose statement or proof uses this declaration.
- IsLocalHomeomorphOn.discreteTopology_of_imageproof · cited by 3
- AlgebraicGeometry.isLocallyArtinian_iff_openCoverproof · cited by 2
- AddAction.IsPretransitive.discreteTopology_iffproof · cited by 2
- MulAction.IsPretransitive.discreteTopology_iffproof · cited by 2
- ChartedSpace.discreteTopologyproof · cited by 1
- AlgebraicGeometry.Scheme.Hom.quasiFiniteLocus_eq_top_iffproof · cited by 1
- Continuous.discrete_of_tendsto_cofinite_cocompactproof · cited by 1
- IsCoveringMapOn.isLocalHomeomorphOnproof · cited by 1