Theorems · Theorem · general topology
isCompact_closure_singleton
∀ {X : Type u_1} [inst : TopologicalSpace X] [R0Space X] {x : X}, IsCompact (closure {x})In an R₀ space, the closure of a singleton is a compact set.
- Defined in
- Mathlib.Topology.Separation.Basic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 77 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpaceR0Space
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites15
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
- Set.iUnionproof · cited by 2,483
- IsOpenproof · cited by 2,400
- IsCompactstatement · cited by 1,282
- closurestatement and proof · cited by 1,254
- Set.iUnion_congr_Propproof · cited by 374
- subset_closureproof · cited by 309
- Set.mem_iUnionproof · cited by 212
- Specializes.mem_openproof · cited by 27
- Set.iUnion_iUnion_eq_leftproof · cited by 27
- R0Spacestatement and proof · cited by 22
Cited by1
Results whose statement or proof uses this declaration.
- Filter.coclosedCompact_le_cofiniteproof · cited by 0