Theorems · Theorem · general topology
closure_Ioc
∀ {α : Type u_1} [inst : TopologicalSpace α] [inst_1 : LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α},
a ≠ b → closure (Set.Ioc a b) = Set.Icc a bThe closure of the interval (a, b] is the closed interval [a, b].
- Defined in
- Mathlib.Topology.Order.DenselyOrdered
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 83 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites17
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
- le_reflproof · cited by 2,061
- Set.Iccstatement and proof · cited by 1,702
- OrderTopologystatement and proof · cited by 1,355
- closurestatement · cited by 1,254
- Set.Iocstatement · cited by 971
- DenselyOrderedstatement and proof · cited by 471
- Set.Subset.transproof · cited by 218
- Set.Subset.antisymmproof · cited by 213
- closure_monoproof · cited by 133
Cited by5
Results whose statement or proof uses this declaration.
- intervalIntegral.sub_le_integral_of_hasDeriv_right_of_leproof · cited by 2
- BoxIntegral.Box.le_TFAEproof · cited by 2
- isClosed_Ioc_iffproof · cited by 1
- frontier_Iocproof · cited by 0
- closure_uIocproof · cited by 0