Theorems · Theorem · general topology
Continuous.isClosedMap
∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] [CompactSpace X] [T2Space Y]
{f : X → Y}, Continuous f → IsClosedMap fA continuous map from a compact space to a Hausdorff space is a closed map.
- Defined in
- Mathlib.Topology.Separation.Hausdorff
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 79 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setproof · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- Continuousstatement and proof · cited by 2,592
- IsClosedproof · cited by 1,639
- T2Spacestatement and proof · cited by 1,351
- CompactSpacestatement and proof · cited by 593
- IsClosedMapstatement · cited by 138
- IsCompact.imageproof · cited by 105
- IsCompact.isClosedproof · cited by 77
- IsClosed.isCompactproof · cited by 43
Cited by9
Results whose statement or proof uses this declaration.
- Continuous.isClosedEmbeddingproof · cited by 5
- Profinite.NobelingProof.isClosed_projproof · cited by 4
- isHomeomorph_iff_continuous_bijectiveproof · cited by 2
- exists_idempotent_of_compact_t2_of_continuous_add_leftproof · cited by 2
- exists_idempotent_of_compact_t2_of_continuous_mul_leftproof · cited by 2
- IsEvenlyCovered.of_openPartialHomeomorphproof · cited by 1
- Continuous.isProperMapproof · cited by 1
- Topology.IsQuotientMap.of_surjective_continuousproof · cited by 0
- ProfiniteGrp.toLimit_surjectiveproof · cited by 0