Theorems · Theorem · general topology
Filter.mem_coclosedCompact_iff
∀ {X : Type u} [inst : TopologicalSpace X] {s : Set X}, s ∈ Filter.coclosedCompact X ↔ IsCompact (closure sᶜ)A set belongs to coclosedCompact if and only if the closure of its complement is compact.
- Defined in
- Mathlib.Topology.Compactness.Compact
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 86 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.
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
- Filterstatement · cited by 8,121
- Compl.complstatement and proof · cited by 2,925
- IsClosedproof · cited by 1,639
- IsCompactstatement and proof · cited by 1,282
- closurestatement and proof · cited by 1,254
- subset_closureproof · cited by 309
- isClosed_closureproof · cited by 195
- Filter.HasBasis.mem_iffproof · cited by 193
- closure_minimalproof · cited by 94
- IsCompact.of_isClosed_subsetproof · cited by 67
Cited by3
Results whose statement or proof uses this declaration.
- hasCompactSupport_iff_eventuallyEqproof · cited by 7
- Filter.compl_mem_coclosedCompactproof · cited by 2
- hasCompactMulSupport_iff_eventuallyEqproof · cited by 1