Theorems · Theorem · general topology
IsProperMap.clusterPt_of_mapClusterPt
∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {f : X → Y},
IsProperMap f → ∀ ⦃ℱ : Filter X⦄ ⦃y : Y⦄, MapClusterPt y ℱ f → ∃ x, f x = y ∧ ClusterPt x ℱBy definition, if f is a proper map and ℱ is any filter on X, then any cluster point of
map f ℱ is the image by f of some cluster point of ℱ.
- Defined in
- Mathlib.Topology.Maps.Proper.Basic
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 50 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
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
- Filterstatement · cited by 8,121
- ClusterPtstatement · cited by 138
- MapClusterPtstatement · cited by 78
- IsProperMapstatement and proof · cited by 66
Cited by4
Results whose statement or proof uses this declaration.
- IsProperMap.compproof · cited by 15
- IsProperMap.isClosedMapproof · cited by 14
- isProperMap_of_comp_of_surjproof · cited by 2
- isProperMap_of_comp_of_injproof · cited by 1