Theorems · Theorem · general topology
Continuous.isClosedEmbedding
∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] [CompactSpace X] [T2Space Y]
{f : X → Y}, Continuous f → Function.Injective f → Topology.IsClosedEmbedding fA continuous injective map from a compact space to a Hausdorff space is a closed embedding.
- Defined in
- Mathlib.Topology.Separation.Hausdorff
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 80 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- TopologicalSpacestatement and proof · cited by 24,529
- Continuousstatement and proof · cited by 2,592
- T2Spacestatement and proof · cited by 1,351
- CompactSpacestatement and proof · cited by 593
- Topology.IsClosedEmbeddingstatement · cited by 195
- Continuous.isClosedMapproof · cited by 9
- Topology.IsClosedEmbedding.of_continuous_injective_isClosedMapproof · cited by 7
Cited by5
Results whose statement or proof uses this declaration.
- Profinite.exists_lift_of_finite_of_injective_of_surjectiveproof · cited by 1
- exists_nat_bool_continuous_surjective_of_compactproof · cited by 0
- exists_embedding_euclidean_of_compactproof · cited by 0
- ENat.isClosedEmbedding_toENNRealproof · cited by 0
- Profinite.Nobeling.isClosedEmbeddingproof · cited by 0