Theorems · Theorem · general topology
IsCompact.diff
∀ {X : Type u} [inst : TopologicalSpace X] {s t : Set X}, IsCompact s → IsOpen t → IsCompact (s \ t)The set difference of a compact set and an open set is a compact set.
- Defined in
- Mathlib.Topology.Compactness.Compact
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 77 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
- IsOpenstatement and proof · cited by 2,400
- IsCompactstatement and proof · cited by 1,282
- isClosed_compl_iffproof · cited by 35
- IsCompact.inter_rightproof · cited by 30
Cited by8
Results whose statement or proof uses this declaration.
- refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_setproof · cited by 3
- IsCompact.everywherePosSubsetproof · cited by 2
- exists_mem_nhds_isCompact_mapsTo_of_isCompact_mem_nhdsproof · cited by 1
- IsCompact.binary_compact_coverproof · cited by 1
- LocallyLipschitzOn.exists_lipschitzOnWith_of_compactproof · cited by 1
- tangentConeAt_nonempty_of_properSpaceproof · cited by 0