Connect RealSet to sage.manifolds

[mkoeppe]

Instances of RealSet that are finite unions of open intervals should be connected in some way to sage.manifolds, in particular to OpenInterval and RealLine (which are both in the global namespace).

In this ticket, we add functionality to initialize a RealSet from instances of OpenInterval and RealLine, as well as from their topological closures (#31644)

[mkoeppe]

Setting new milestone based on a cursory review of ticket status, priority, and last modification date.

[mkoeppe]

Last 10 new commits:

​e7f7a7d	Merge #31764
​9259a2c	Merge #31798
​ac53984	src/sage/manifolds/subsets/__init__.py: New
​21739de	src/sage/manifolds/subset.py: Simplify code, make pyflakes happy
​6bbd4ae	src/sage/manifolds/subset.py: Add coding header
​2d7fda7	src/sage/manifolds/subset.py: Simplify code more to make pyflakes happy
​8d3ad85	Merge #31764
​60505b0	Merge #31798
​9dd9f90	Merge #31644
​fb33c0f	RealSet: Allow initialization from RealLine and OpenInterval instances and OpenInterval closures

[gh-mjungmath]

Restricted to open intervals (not closures), this is nothing but the forgetful functor from manifolds into sets. Perhaps it is a good idea to add this functor somehow?

Other than that it looks good! Let's wait for the patchbot.

[mkoeppe]

... or rather from manifolds to topological spaces (see #31877).

Do we have forgetful functors anywhere already?

[mkoeppe]

We may also want to consider

• adding a method RealSet.as_manifold_subset
• extending the RealSet constructors (default constructor and RealSet.open etc.) so that they can build manifold objects; for example if structure or name is passed; then perhaps the global bindings RealLine and OpenInterval can be deprecated.

[gh-mjungmath]

Notice that RealSet supports unions of intervals whereas OpenInterval does not.

[gh-mjungmath]

Moreover, you represented closed or clopen intervals by a topological closure, which is fine for now. But as soon as we have manifolds with boundary, we probably want them to be represented by those objects/instances instead (or as well).

[gh-mjungmath]

LGTM

[mkoeppe]

Replying to gh-mjungmath:

Notice that RealSet supports unions of intervals whereas OpenInterval does not.

Yes, it does. You can initialize a RealSet using several OpenInterval instances, and it will take the union.

[mkoeppe]

Thanks for the review!

[gh-mjungmath]

Replying to mkoeppe:

Replying to gh-mjungmath:

Notice that RealSet supports unions of intervals whereas OpenInterval does not.

Yes, it does. You can initialize a RealSet using several OpenInterval instances, and it will take the union.

That is not what I mean. There are no unions of OpenInterval on the manifold level yet. This is, for example, important for a potential RealSet.as_manifold_subset.

[mkoeppe]

Replying to gh-mjungmath:

There are no unions of OpenInterval on the manifold level yet. This is, for example, important for a potential RealSet.as_manifold_subset.

If you create OpenInterval instances with the same ambient_interval set, then you can take unions.

sage: I02 = OpenInterval(0, 2, ambient_interval=RealLine())
sage: I13 = OpenInterval(1, 3, ambient_interval=RealLine())
sage: I02.union(I13)
Open subset (0, 2)_union_(1, 3) of the Real number line R

[gh-mjungmath]

Right, but

sage: I02 = OpenInterval(0, 2, ambient_interval=RealLine())
sage: I13 = OpenInterval(1, 3, ambient_interval=RealLine())
sage: U = I02.union(I13)
sage: RealSet(U)

won't work, no?

[mkoeppe]

Good catch...

[mkoeppe]

Folllow up in #31878.

[tscrim]

Replying to mkoeppe:

... or rather from manifolds to topological spaces (see #31877).

Do we have forgetful functors anywhere already?

categories.functor.ForgetfulFunctor

[git]

Branch pushed to git repo; I updated commit sha1 and set ticket back to needs_review. Last 10 new commits:

​fd4506a	Merge #31732
​7fed9ef	Merge #31736
​c4acd09	Merge tag '9.4.beta2' into t/31763/manifoldsubset__new_methods_declare_subset__declare_superset
​2ba18ab	Merge #31763
​0e9e5cd	Merge #31732
​a96f736	Merge #31736
​582b58d	Merge #31763
​2113f78	Merge #31764
​9abc617	Merge #31798
​785aed0	Merge #31644

[mkoeppe]

Trivial merge with updated dependency

[git]

Branch pushed to git repo; I updated commit sha1 and set ticket back to needs_review. New commits:

​4b09685	RealSet: Put it in a suitable subcategory of TopologicalSpaces()
​46eed0e	RealSet.ambient: Change to a normal method
​5b8ceca	InternalRealInterval, RealSet: Add _latex_ methods
​69ca854	RealSet._repr_: Use unicode cup sign instead of +
​dbdfc06	InternalRealInterval, RealSet: Remove extra whitespace in latex, add documentation
​7f56338	PiecewiseFunction: Adjust doctests for changed RealSet repr
​8abdc8b	src/sage/functions/piecewise.py: Add coding header
​5b0f85d	Merge #31880
​81b43f4	Merge branch 't/31877/refine_category_of_realset' into t/30832/connect_realset_to_sage_manifolds

[mkoeppe]

Another trivial merge to remove a merge conflict