Theorems · Theorem · general topology
mem_closure_iff_clusterPt
∀ {X : Type u} [inst : TopologicalSpace X] {x : X} {s : Set X}, x ∈ closure s ↔ ClusterPt x (Filter.principal s)- Defined in
- Mathlib.Topology.ClusterPt
- Cited by
- 16 results in Mathlib
- Foundations
- Depth 73 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpace
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.
- Setstatement and proof · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- closurestatement · cited by 1,254
- Filter.principalstatement · cited by 740
- ClusterPtstatement · cited by 138
- mem_closure_iff_frequentlyproof · cited by 15
- clusterPt_principal_iff_frequentlyproof · cited by 2
Cited by16
Results whose statement or proof uses this declaration.
- mem_closure_iff_nhdsproof · cited by 20
- mem_closure_iff_nhdsWithin_neBotproof · cited by 16
- specializes_TFAEproof · cited by 6
- closure_eq_cluster_ptsproof · cited by 5
- uniqueDiffWithinAt_iff_accPtproof · cited by 5
- Set.EqOn.of_subset_closureproof · cited by 4
- mem_closure_iff_nhds_basis'proof · cited by 3
- ClusterPt.mem_closureproof · cited by 3
- mem_closure_iff_nhds_ne_botproof · cited by 2
- IsPreconnected.preperfect_of_nontrivialproof · cited by 2
- DenseRange.nhdsWithin_neBotproof · cited by 2
- Subalgebra.frontier_spectrumproof · cited by 2