Theorems · Theorem · general topology
discreteTopology_iff_forall_isClosed
∀ {α : Type u_1} [inst : TopologicalSpace α], DiscreteTopology α ↔ ∀ (s : Set α), IsClosed s- Defined in
- Mathlib.Topology.Order
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 67 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
- IsClosedstatement · cited by 1,639
- DiscreteTopologystatement · cited by 373
- Function.Surjective.forallproof · cited by 214
- isOpen_compl_iffproof · cited by 63
- compl_surjectiveproof · cited by 26
- discreteTopology_iff_forall_isOpenproof · cited by 3
Cited by2
Results whose statement or proof uses this declaration.
- PrimeSpectrum.discreteTopology_iff_finite_and_krullDimLE_zeroproof · cited by 3
- DiscreteTopology.of_finite_of_isClosed_singletonproof · cited by 1