Theorems · Theorem · general topology

nhds_basis_Ioo_pos_of_pos

∀ {α : Type u_1} [inst : TopologicalSpace α] [inst_1 : AddCommGroup α] [inst_2 : LinearOrder α] [IsOrderedAddMonoid α]
  [OrderTopology α] [NoMaxOrder α] {a : α},
  0 < a → (nhds a).HasBasis (fun ε => 0 < ε ∧ ε ≤ a) fun ε => Set.Ioo (a - ε) (a + ε)

If a is positive, then the intervals (a - ε, a + ε), 0 < ε ≤ a, form a basis of neighborhoods of a. This upper bound for ε guarantees that all elements of these intervals are positive.

Defined in

Mathlib.Topology.Order.LeftRightNhds

Cited by

0 results in Mathlib

Foundations

Depth 68 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceAddCommGroupLinearOrderIsOrderedAddMonoidOrderTopologyNoMaxOrder