Theorems · Definition · general topology

Topology.IsEmbedding.toHomeomorph

{X : Type u_1} →
  {Y : Type u_2} →
    [inst : TopologicalSpace X] →
      [inst_1 : TopologicalSpace Y] → {f : X → Y} → Topology.IsEmbedding f → X ≃ₜ ↑(Set.range f)

Homeomorphism given an embedding.

Defined in: Mathlib.Topology.Homeomorph.Lemmas
Cited by: 16 results in Mathlib
Foundations: Depth 78 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: TopologicalSpace TopologicalSpace

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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.Elemstatement · cited by 7,166
Set.rangestatement · cited by 4,705
Homeomorphstatement · cited by 725
Topology.IsEmbeddingstatement and proof · cited by 294
Topology.IsEmbedding.injectiveproof · cited by 103
Equiv.ofInjectiveproof · cited by 64
Equiv.toHomeomorphOfIsInducingproof · cited by 8

Cited by22

Results whose statement or proof uses this declaration.

AlgebraicGeometry.Scheme.Hom.fiberHomeoproof · cited by 6
FiberBundle.homeomorphAtproof · cited by 4
Topology.IsEmbedding.homeomorphImageproof · cited by 3
Topology.IsEmbedding.measurableEmbeddingproof · cited by 3
AlgebraicGeometry.LocallyOfFiniteType.jacobsonSpaceproof · cited by 2
ContinuousMap.exists_extensionproof · cited by 2
AlgebraicGeometry.quasiCompact_affineProperty_iff_quasiSeparatedSpaceproof · cited by 1
isEmbedding_of_iSup_eq_top_of_preimage_subset_rangeproof · cited by 1
Topology.IsEmbedding.toHomeomorph_apply_coestatement and proof · cited by 1
unitsHomeomorphNeZeroproof · cited by 1
TopCat.binaryCofan_isColimit_iffproof · cited by 1
TopCat.snd_iso_of_left_embedding_range_subsetproof · cited by 1