Theorems · Theorem · general topology
discreteTopology_iff_forall_isOpen
∀ {α : Type u_1} [inst : TopologicalSpace α], DiscreteTopology α ↔ ∀ (s : Set α), IsOpen s- Defined in
- Mathlib.Topology.Order
- Cited by
- 3 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 and proof · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- IsOpenstatement and proof · cited by 2,400
- DiscreteTopologystatement · cited by 373
- isOpen_discreteproof · cited by 36
- eq_bot_of_singletons_openproof · cited by 4
Cited by3
Results whose statement or proof uses this declaration.
- discreteTopology_iff_forall_isClosedproof · cited by 2
- DiscreteTopology.of_continuous_injectiveproof · cited by 2
- JacobsonSpace.discreteTopologyproof · cited by 0