Theorems · Theorem · general topology
IsComplete.isClosed
∀ {α : Type u_1} [inst : UniformSpace α] [T0Space α] {s : Set α}, IsComplete s → IsClosed sIn a separated space, a complete set is closed.
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 92 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- UniformSpaceT0Space
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites17
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
- Filterproof · cited by 8,121
- nhdsproof · cited by 5,554
- UniformSpacestatement and proof · cited by 2,040
- nhdsWithinproof · cited by 1,912
- IsClosedstatement · cited by 1,639
- Filter.principalproof · cited by 740
- inf_le_leftproof · cited by 286
- inf_le_rightproof · cited by 238
- T0Spacestatement and proof · cited by 179
- ClusterPtproof · cited by 138
- Cauchyproof · cited by 115
Cited by6
Results whose statement or proof uses this declaration.
- Submodule.closed_of_finiteDimensionalproof · cited by 8
- IsUniformEmbedding.isClosedEmbeddingproof · cited by 7
- AntilipschitzWith.isClosed_rangeproof · cited by 4
- IsCompactOperator.restrict'proof · cited by 2
- OrthogonalFamily.range_linearIsometryproof · cited by 1
- MeasureTheory.isClosed_aestronglyMeasurableproof · cited by 1