Theorems · Theorem · general topology
Topology.IsOpenEmbedding.isOpen_iff_image_isOpen
∀ {X : Type u_1} {Y : Type u_2} {f : X → Y} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y],
Topology.IsOpenEmbedding f → ∀ {s : Set X}, IsOpen s ↔ IsOpen (f '' s)- Defined in
- Mathlib.Topology.Maps.Basic
- Cited by
- 7 results in Mathlib
- Foundations
- Depth 77 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
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.imagestatement and proof · cited by 5,609
- Set.preimageproof · cited by 4,946
- IsOpenstatement and proof · cited by 2,400
- Topology.IsOpenEmbeddingstatement and proof · cited by 231
- IsOpen.preimageproof · cited by 147
- Topology.IsEmbedding.injectiveproof · cited by 103
- Set.preimage_image_eqproof · cited by 87
- Topology.IsOpenEmbedding.toIsEmbeddingproof · cited by 61
- Topology.IsEmbedding.continuousproof · cited by 51
- Topology.IsOpenEmbedding.isOpenMapproof · cited by 50
Cited by7
Results whose statement or proof uses this declaration.
- TopologicalSpace.IsOpenCover.isOpen_iff_coe_preimageproof · cited by 4
- AlgebraicGeometry.Scheme.Hom.quasiFiniteAt_iff_isOpen_singleton_asFiberproof · cited by 2
- Topology.IsOpenEmbedding.prespectralSpaceproof · cited by 1
- Topology.IsCoinducing.restrictPreimage_of_isOpenproof · cited by 1
- MeasureTheory.Measure.Regular.comap'proof · cited by 1
- MeasureTheory.Measure.IsOpenPosMeasure.comapproof · cited by 0
- Topology.IsOpenEmbedding.isOpen_iff_preimage_isOpenproof · cited by 0