Theorems · Theorem · general topology
accPt_principal_iff_clusterPt
∀ {X : Type u} [inst : TopologicalSpace X] {x : X} {C : Set X},
AccPt x (Filter.principal C) ↔ ClusterPt x (Filter.principal (C \ {x}))x is an accumulation point of a set C iff it is a cluster point of C ∖ {x}.
- Defined in
- Mathlib.Topology.ClusterPt
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 52 from the axioms · uses propext, Quot.sound
- Assumes
- TopologicalSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
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
- Filterproof · cited by 8,121
- Compl.complproof · cited by 2,925
- Filter.principalstatement and proof · cited by 740
- Set.inter_commproof · cited by 291
- ClusterPtstatement and proof · cited by 138
- AccPtstatement · cited by 75
- Set.sdiff_eqproof · cited by 59
- Filter.inf_principalproof · cited by 29
- accPt_iff_clusterPtproof · cited by 7
Cited by4
Results whose statement or proof uses this declaration.
- accPt_principal_iff_nhdsWithinproof · cited by 6
- isClosed_iff_derivedSet_subsetproof · cited by 3
- IsPreconnected.preperfect_of_nontrivialproof · cited by 2
- clusterPt_principalproof · cited by 0