Theorems · Theorem · general topology
IsCompact.bddBelow
∀ {α : Type u_2} [inst : LinearOrder α] [inst_1 : TopologicalSpace α] [ClosedIicTopology α] [Nonempty α] {s : Set α},
IsCompact s → BddBelow sA compact set is bounded below
- Defined in
- Mathlib.Topology.Order.Compact
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 79 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
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.Nonemptyproof · cited by 2,627
- IsCompactstatement and proof · cited by 1,282
- BddBelowstatement · cited by 401
- Set.eq_empty_or_nonemptyproof · cited by 248
- lowerBoundsproof · cited by 212
- IsLeastproof · cited by 122
- ClosedIicTopologystatement and proof · cited by 115
- IsCompact.exists_isLeastproof · cited by 9
- bddBelow_emptyproof · cited by 5
Cited by8
Results whose statement or proof uses this declaration.
- IsCompact.bddAboveproof · cited by 6
- IsCompact.isGLB_sInfproof · cited by 3
- IsCompact.bddBelow_imageproof · cited by 2
- locallyIntegrableOn_mul_sum_Iccproof · cited by 2
- IsCompact.exists_sInf_image_eq_and_leproof · cited by 2
- Continuous.bddBelow_range_of_hasCompactSupportproof · cited by 1
- eq_Icc_of_connected_compactproof · cited by 1
- Continuous.bddBelow_range_of_hasCompactMulSupportproof · cited by 1