Theorems · Theorem · general topology
specializingMap_iff_closure_singleton
∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {f : X → Y},
Continuous f → (SpecializingMap f ↔ ∀ (x : X), f '' closure {x} = closure {f x})- Defined in
- Mathlib.Topology.Inseparable
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 80 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- Set.imagestatement and proof · cited by 5,609
- Continuousstatement and proof · cited by 2,592
- closurestatement · cited by 1,254
- SpecializingMapstatement and proof · cited by 21
- Continuous.specialization_monotoneproof · cited by 3
- closure_singleton_eq_Iicproof · cited by 3
- Relation.fibration_iff_image_Iicproof · cited by 2
Cited by1
Results whose statement or proof uses this declaration.
- specializingMap_iff_isClosed_image_closure_singletonproof · cited by 0