Theorems · Theorem · general topology
specializes_iff_forall_closed
∀ {X : Type u_1} [inst : TopologicalSpace X] {x y : X}, x ⤳ y ↔ ∀ (s : Set X), IsClosed s → x ∈ s → y ∈ s- Defined in
- Mathlib.Topology.Inseparable
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 75 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.
Cites10
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
- nhdsproof · cited by 5,554
- IsOpenproof · cited by 2,400
- IsClosedstatement and proof · cited by 1,639
- closureproof · cited by 1,254
- List.TFAE.outproof · cited by 177
- Specializesstatement and proof · cited by 176
- ClusterPtproof · cited by 138
- specializes_TFAEproof · cited by 6
Cited by2
Results whose statement or proof uses this declaration.
- Specializes.mem_closedproof · cited by 8
- not_specializes_iff_exists_closedproof · cited by 0