Theorems · Theorem · general topology
Topology.IsOpenEmbedding.image_mem_nhds
∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {f : X → Y},
Topology.IsOpenEmbedding f → ∀ {s : Set X} {x : X}, f '' s ∈ nhds (f x) ↔ s ∈ nhds x- Defined in
- Mathlib.Topology.Maps.Basic
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 73 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
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
- Filterstatement and proof · cited by 8,121
- Set.imagestatement and proof · cited by 5,609
- nhdsstatement and proof · cited by 5,554
- Topology.IsOpenEmbeddingstatement and proof · cited by 231
- Topology.IsEmbedding.injectiveproof · cited by 103
- Set.preimage_image_eqproof · cited by 87
- Filter.mem_mapproof · cited by 63
- Topology.IsOpenEmbedding.toIsEmbeddingproof · cited by 61
- Topology.IsOpenEmbedding.map_nhds_eqproof · cited by 22
Cited by4
Results whose statement or proof uses this declaration.
- smul_mem_nhds_smul_iffproof · cited by 5
- vadd_mem_nhds_vadd_iffproof · cited by 5
- Topology.IsLocallyConstructible.of_isOpenCoverproof · cited by 3
- UpperHalfPlane.isOpenMap_normproof · cited by 1