Theorems · Theorem · general topology
SeparatedNhds.of_isClosed_isCompact_closure_compl_isClosed
∀ {X : Type u_1} [inst : TopologicalSpace X] [R1Space X] {s t : Set X},
IsClosed s → IsCompact (closure sᶜ) → IsClosed t → Disjoint s t → SeparatedNhds s tIn a R1Space X, for disjoint closed sets s t such that closure sᶜ is compact,
there are neighbourhoods that separate s and t.
- Defined in
- Mathlib.Topology.Separation.Hausdorff
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 84 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpaceR1Space
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites31
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
- LE.le.transproof · cited by 3,151
- Compl.complstatement and proof · cited by 2,925
- Disjointstatement and proof · cited by 2,201
- IsClosedstatement and proof · cited by 1,639
- le_rflproof · cited by 1,558
- IsCompactstatement and proof · cited by 1,282
- closurestatement and proof · cited by 1,254
- interiorproof · cited by 714
- Set.inter_subset_leftproof · cited by 360
- Set.inter_subset_rightproof · cited by 329
Cited by1
Results whose statement or proof uses this declaration.
- exists_tsupport_one_of_isOpen_isClosedproof · cited by 2