Theorems · Theorem · general topology
isHomeomorph_iff_isQuotientMap_injective
∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {f : X → Y},
IsHomeomorph f ↔ Topology.IsQuotientMap f ∧ Function.Injective fA map is a homeomorphism iff it is a quotient map and injective.
- Defined in
- Mathlib.Topology.Homeomorph.Lemmas
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 73 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites12
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
- IsOpenproof · cited by 2,400
- Topology.IsQuotientMapstatement and proof · cited by 124
- Set.preimage_image_eqproof · cited by 87
- IsHomeomorphstatement and proof · cited by 69
- Topology.IsQuotientMap.isCoinducingproof · cited by 35
- Topology.IsQuotientMap.surjectiveproof · cited by 24
- Topology.IsCoinducing.isOpen_preimageproof · cited by 15
- Topology.IsQuotientMap.continuousproof · cited by 14
- IsHomeomorph.injectiveproof · cited by 2
- IsHomeomorph.isQuotientMapproof · cited by 2
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