chore(topology): add 2 missing inhabited instances (#10446)

Also add an instance from `discrete_topology` to `topological_ring`.

src/topology/algebra/ring.lean

@@ -39,6 +39,10 @@ If `R` is a ring, then negation is automatically continuous, as it is multiplica
 class topological_ring [topological_space α] [semiring α]
   extends has_continuous_add α, has_continuous_mul α : Prop
 
+@[priority 50]
+instance discrete_topology.topological_ring {α} [topological_space α] [semiring α]
+  [discrete_topology α] : topological_ring α := ⟨⟩
+
 section
 variables {α} [topological_space α] [semiring α] [topological_ring α]
 

src/topology/category/Top/open_nhds.lean

@@ -60,6 +60,8 @@ instance (x : X) : order_top (open_nhds x) :=
 { top := ⟨⊤, trivial⟩,
   le_top := λ _, le_top }
 
+instance (x : X) : inhabited (open_nhds x) := ⟨⊤⟩
+
 instance open_nhds_category (x : X) : category.{u} (open_nhds x) :=
 by {unfold open_nhds, apply_instance}
 

src/topology/category/TopCommRing.lean

@@ -27,6 +27,8 @@ structure TopCommRing :=
 
 namespace TopCommRing
 
+instance : inhabited TopCommRing := ⟨⟨punit⟩⟩
+
 instance : has_coe_to_sort TopCommRing (Type u) := ⟨TopCommRing.α⟩
 
 attribute [instance] is_comm_ring is_topological_space is_topological_ring