Theorems · Definition · general topology

Topology.IsEmbedding.homeomorphImage

{X : Type u_1} →
  {Y : Type u_2} →
    [inst : TopologicalSpace X] →
      [inst_1 : TopologicalSpace Y] → {f : X → Y} → Topology.IsEmbedding f → (s : Set X) → ↑s ≃ₜ ↑(f '' s)

A set is homeomorphic to its image under any embedding.

Defined in: Mathlib.Topology.Homeomorph.Lemmas
Cited by: 3 results in Mathlib
Foundations: Depth 79 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: TopologicalSpace TopologicalSpace

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Cites9

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.Elemstatement · cited by 7,166
Set.imagestatement · cited by 5,609
Homeomorphstatement · cited by 725
Topology.IsEmbeddingstatement and proof · cited by 294
Homeomorph.transproof · cited by 49
Homeomorph.setCongrproof · cited by 29
Topology.IsEmbedding.toHomeomorphproof · cited by 16

Cited by4

Results whose statement or proof uses this declaration.

Topology.IsEmbedding.homeomorphOfSubsetRangeproof · cited by 2
IsEvenlyCovered.comp_subtypeValproof · cited by 1
Topology.IsEmbedding.isSimplyConnected_imageproof · cited by 1
IsEvenlyCovered.subtypeVal_compproof · cited by 1