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__.

Depends on #30799

CC: @egourgoulhon @tscrim @tobiasdiez

Component: manifolds

Author: Michael Jung

Branch/Commit: 200942c

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/30830