Theorems · Theorem · general topology
specializes_iff_mem_closure
∀ {X : Type u_1} [inst : TopologicalSpace X] {x y : X}, x ⤳ y ↔ y ∈ closure {x}- Defined in
- Mathlib.Topology.Inseparable
- Cited by
- 11 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
- IsClosedproof · cited by 1,639
- closurestatement and proof · 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 by11
Results whose statement or proof uses this declaration.
- PrimeSpectrum.le_iff_specializesproof · cited by 8
- closure_singleton_eq_Iicproof · cited by 3
- addGroup_inseparable_iffproof · cited by 1
- isCompact_closure_singletonproof · cited by 1
- isClosed_iUnion_closure_singleton_of_not_tendstoproof · cited by 1
- norm_image_of_norm_eq_zeroproof · cited by 1
- Seminorm.map_eq_zero_of_norm_eq_zeroproof · cited by 1
- Specializes.mem_closureproof · cited by 1
- isPreirreducible_iff_forall_mem_subset_closure_singletonproof · cited by 0
- TopologicalSpace.vietoris.specializes_iffproof · cited by 0
- group_inseparable_iffproof · cited by 0