Theorems · Theorem · general topology
isClosed_iff_clusterPt
∀ {X : Type u} [inst : TopologicalSpace X] {s : Set X}, IsClosed s ↔ ∀ (a : X), ClusterPt a (Filter.principal s) → a ∈ s- Defined in
- Mathlib.Topology.ClusterPt
- Cited by
- 7 results in Mathlib
- Foundations
- Depth 75 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.
Cites6
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
- IsClosedstatement · cited by 1,639
- Filter.principalstatement and proof · cited by 740
- ClusterPtstatement and proof · cited by 138
- closure_subset_iff_isClosedproof · cited by 11
Cited by7
Results whose statement or proof uses this declaration.
- IsClosed.isCompleteproof · cited by 19
- IsComplete.isClosedproof · cited by 6
- isClosed_iff_derivedSet_subsetproof · cited by 3
- isClosed_and_discrete_iffproof · cited by 2
- exists_nat_nat_continuous_surjective_of_completeSpaceproof · cited by 1
- IsCountablyCompact.of_isClosed_subsetproof · cited by 1
- ZeroAtInftyContinuousMap.isClosed_range_toBCFproof · cited by 0