Theorems · Theorem · general topology
IsCompact.exists_isLeast
∀ {α : Type u_2} [inst : LinearOrder α] [inst_1 : TopologicalSpace α] [ClosedIicTopology α] {s : Set α},
IsCompact s → s.Nonempty → ∃ x, IsLeast s x- Defined in
- Mathlib.Topology.Order.Compact
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 78 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites19
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
- LinearOrderstatement and proof · cited by 8,572
- Set.Elemproof · cited by 7,166
- Set.Nonemptystatement and proof · cited by 2,627
- le_rflproof · cited by 1,558
- IsCompactstatement and proof · cited by 1,282
- Set.Iicproof · cited by 1,111
- Set.iInterproof · cited by 1,084
- IsLeaststatement · cited by 122
- ClosedIicTopologystatement and proof · cited by 115
- Set.not_nonempty_iff_eq_emptyproof · cited by 56
Cited by9
Results whose statement or proof uses this declaration.
- IsCompact.exists_isMinOnproof · cited by 15
- IsCompact.bddBelowproof · cited by 8
- IsCompact.sInf_memproof · cited by 5
- atBot_le_cocompactproof · cited by 4
- IsCompact.exists_isGreatestproof · cited by 2
- Unitary.norm_sub_one_lt_two_iffproof · cited by 1
- Continuous.exists_forall_le_of_hasCompactMulSupportproof · cited by 1
- Continuous.exists_forall_le_of_hasCompactSupportproof · cited by 1
- IsCompact.exists_isGLBproof · cited by 1