Theorems · Theorem · general topology
closure_Ioi
∀ {α : Type u_1} [inst : TopologicalSpace α] [inst_1 : LinearOrder α] [OrderTopology α] [DenselyOrdered α] (a : α)
[NoMaxOrder α], closure (Set.Ioi a) = Set.Ici aThe closure of the interval (a, +∞) is the closed interval [a, +∞).
- Defined in
- Mathlib.Topology.Order.DenselyOrdered
- Cited by
- 11 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.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- LinearOrderstatement and proof · cited by 8,572
- Set.Ioistatement · cited by 1,463
- OrderTopologystatement and proof · cited by 1,355
- closurestatement · cited by 1,254
- Set.Icistatement · cited by 1,070
- DenselyOrderedstatement and proof · cited by 471
- NoMaxOrderstatement and proof · cited by 340
- Set.nonempty_Ioiproof · cited by 10
- closure_Ioi'proof · cited by 5
Cited by11
Results whose statement or proof uses this declaration.
- uniqueDiffWithinAt_Ioiproof · cited by 5
- PhragmenLindelof.quadrant_Iproof · cited by 4
- Complex.closure_setOfPred_lt_reproof · cited by 3
- image_le_of_liminf_slope_right_le_deriv_boundaryproof · cited by 2
- stronglyMeasurable_derivWithin_Iciproof · cited by 2
- Complex.closure_setOfPred_lt_improof · cited by 2
- not_differentiableWithinAt_of_deriv_tendsto_atTop_Ioiproof · cited by 2
- le_gronwallBound_of_liminf_deriv_right_leproof · cited by 1
- closure_of_rat_image_ltproof · cited by 0
- closure_open_halfSpaceproof · cited by 0
- ConvexCone.Pointed.of_nonempty_of_isClosedproof · cited by 0