Trac #30830: Subintervals of OpenInterval and UniqueRepresentation

At the moment, we have the following behavior:
{{{
sage: I = OpenInterval(0,2)
sage: J = OpenInterval(0,1, ambient_interval=I, coordinate='t')
sage: I.open_interval(0,1)
Traceback (most recent call last)
...
ValueError: the name '(0, 1)' is already used for another subset of the
Real interval (0, 2)
}}}

Even though the use of `OpenInterval(0,1, ambient_interval=I)` is not
intended, this is still a blind spot.

The reason for this behavior comes from the `UniqueRepresentation` and
how the subintervals are constructed.

I propose a fix using `__classcall_private__`.

URL: https://trac.sagemath.org/30830
Reported by: gh-mjungmath
Ticket author(s): Michael Jung
Reviewer(s): Travis Scrimshaw

src/sage/manifolds/differentiable/examples/real_line.py

@@ -290,6 +290,34 @@ class OpenInterval(DifferentiableManifold):
         t: (1/2, 1)
 
     """
+    @staticmethod
+    def __classcall_private__(cls, lower, upper, ambient_interval=None,
+                              name=None, latex_name=None, coordinate=None,
+                              names=None, start_index=0):
+        r"""
+        Determine the correct interval to return based upon the input.
+
+        TESTS:
+
+        Check whether :trac:`30830` is fixed::
+
+            sage: I = OpenInterval(0,2)
+            sage: J = OpenInterval(0,1, ambient_interval=I, coordinate='t')
+            sage: I.open_interval(0,1)
+            Real interval (0, 1)
+
+        """
+        if ambient_interval:
+            # cope the UniqueRepresentation framework for subintervals and
+            # reset irrelevant information only:
+            coordinate = None
+            names = None
+            start_index = 0
+        return super(cls, OpenInterval).__classcall__(cls, lower, upper,
+                          ambient_interval=ambient_interval, name=name,
+                          latex_name=latex_name, coordinate=coordinate,
+                          names=names, start_index=start_index)
+
     def __init__(self, lower, upper, ambient_interval=None,
                  name=None, latex_name=None,
                  coordinate=None, names=None, start_index=0):
@@ -830,6 +858,27 @@ class RealLine(OpenInterval):
         [Real interval (0, 1), Real number line R]
 
     """
+    @staticmethod
+    def __classcall__(cls, name='R', latex_name=r'\Bold{R}', coordinate=None,
+                      names=None, start_index=0):
+        r"""
+        Determine the correct interval to return based upon the input.
+
+        TESTS::
+
+            sage: R = RealLine(); R
+            Real number line R
+            sage: R1 = RealLine('R'); R1
+            Real number line R
+            sage: R is R1
+            True
+
+        """
+        return super(cls, RealLine).__classcall__(cls, name=name,
+                                           latex_name=latex_name,
+                                           coordinate=coordinate,
+                                           names=names, start_index=start_index)
+
     def __init__(self, name='R', latex_name=r'\Bold{R}', coordinate=None,
                  names=None, start_index=0):
         r"""