RealSet: Use ambient.manifold().canonical_chart(), add nested example

src/sage/sets/real_set.py

@@ -890,7 +890,7 @@ def __classcall__(cls, *args, **kwds):
                 from sage.misc.latex import latex
                 latex_name = latex(real_set)
 
-            return ambient.canonical_chart().pullback(real_set, name=name, latex_name=latex_name)
+            return ambient.manifold().canonical_chart().pullback(real_set, name=name, latex_name=latex_name)
 
         from sage.symbolic.expression import Expression
         if len(args) == 1 and isinstance(args[0], RealSet):
@@ -1047,19 +1047,37 @@ def __init__(self, *intervals):
 
             sage: RealSet(-oo, oo, structure='differentiable')
             Real number line ℝ
+
             sage: RealSet([0, 1], structure='differentiable')
             Subset [0, 1] of the Real number line ℝ
             sage: _.category()
             Category of subobjects of sets
+
             sage: RealSet.open_closed(0, 5, structure='differentiable')
             Subset (0, 5] of the Real number line ℝ
+
             sage: RealSet(0, 1, name='A')
             Open subset A of the Real number line ℝ
             sage: _.category()
             Join of
              Category of smooth manifolds over Real Field with 53 bits of precision and
              Category of connected manifolds over Real Field with 53 bits of precision and
              Category of subobjects of sets
+
+            sage: R_xi.<ξ> = RealSet(-oo, oo); R_xi
+            Real number line ℝ
+            sage: R_xi.canonical_chart()
+            Chart (ℝ, (ξ,))
+            sage: P_xi = RealSet(0, oo, ambient=R_xi); P_xi
+            Open subset (0, +oo) of the Real number line ℝ
+            sage: P_xi.default_chart()
+            Chart ((0, +oo), (ξ,))
+            sage: B_xi = RealSet(0, 1, ambient=P_xi); B_xi
+            Open subset (0, 1) of the Real number line ℝ
+            sage: B_xi.default_chart()
+            Chart ((0, 1), (ξ,))
+            sage: R_xi.subset_family()
+            Set {(0, +oo), (0, 1), ℝ} of open subsets of the Real number line ℝ
         """
         category = TopologicalSpaces()
         if len(intervals) <= 1: