Trac #31732: ManifoldSubset: New methods declare_empty, declare_nonem…

…pty, is_empty, has_defined_points, open_cover_family

A subset is declared empty by giving it (or any superset) an open cover
that is an empty family.

A subset is declared nonempty by defining a point on it (or on any
subset).

URL: https://trac.sagemath.org/31732
Reported by: mkoeppe
Ticket author(s): Matthias Koeppe
Reviewer(s): Michael Jung

src/sage/manifolds/point.py

@@ -182,7 +182,10 @@ def __init__(self, parent, coords=None, chart=None, name=None,
             sage: TestSuite(q).run()
 
         """
+        if parent.is_empty():
+            raise TypeError(f'cannot define a point on the {parent} because it has been declared empty')
         Element.__init__(self, parent)
+        parent._has_defined_points = True
         self._manifold = parent.manifold()  # a useful shortcut
         self._coordinates = {} # dictionary of the point coordinates in various
                                # charts, with the charts as keys