Theorems · Theorem · general topology
preimage_closedPoints_subset
∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {f : X → Y},
Function.Injective f → Continuous f → f ⁻¹' closedPoints Y ⊆ closedPoints X- Defined in
- Mathlib.Topology.JacobsonSpace
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 58 from the axioms · uses propext, Classical.choice, Quot.sound
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
- Set.preimagestatement and proof · cited by 4,946
- Continuousstatement and proof · cited by 2,592
- IsClosedproof · cited by 1,639
- Set.image_singletonproof · cited by 174
- Set.preimage_image_eqproof · cited by 87
- closedPointsstatement and proof · cited by 25
- continuous_iff_isClosedproof · cited by 24
- mem_closedPoints_iffproof · cited by 2
Cited by1
Results whose statement or proof uses this declaration.
- Topology.IsOpenEmbedding.preimage_closedPointsproof · cited by 1