Theorems · Theorem · order theory

AddSubgroup.discrete_iff_addCyclic

∀ {G : Type u_1} [inst : AddCommGroup G] [inst_1 : LinearOrder G] [IsOrderedAddMonoid G] [inst_3 : TopologicalSpace G]
  [OrderTopology G] [Archimedean G] {H : AddSubgroup G}, IsAddCyclic ↥H ↔ DiscreteTopology ↥H

In an Archimedean linearly ordered additive group (with the order topology), a subgroup is discrete iff it is cyclic.

Defined in

Mathlib.Topology.Algebra.Order.ArchimedeanDiscrete

Cited by

1 results in Mathlib

Foundations

Depth 91 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddCommGroupLinearOrderIsOrderedAddMonoidTopologicalSpaceOrderTopologyArchimedean