Theorems · Theorem · general topology
eq_bot_of_singletons_open
∀ {α : Type u_1} {t : TopologicalSpace α}, (∀ (x : α), IsOpen {x}) → t = ⊥- Defined in
- Mathlib.Topology.Order
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 64 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
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
- Bot.botstatement · cited by 4,720
- IsOpenstatement and proof · cited by 2,400
- bot_uniqueproof · cited by 57
- Set.biUnion_of_singletonproof · cited by 43
- isOpen_biUnionproof · cited by 35
Cited by4
Results whose statement or proof uses this declaration.
- discreteTopology_iff_isOpen_singletonproof · cited by 8
- discreteTopology_iff_forall_isOpenproof · cited by 3
- induced_topology_pureproof · cited by 1
- exists_topology_isEmbedding_natproof · cited by 1