Topological closure of manifold subsets, methods ManifoldSubset.closure, is_closed, declare_closed

[mkoeppe]

We define a subclass of ManifoldSubset whose instances represents the topological closure of given subset in the manifold.

Subsets provide a method closure to construct it. When the subset is already closed, as detected by the new method is_closed, it just returns the input.

We also add a method declare_closed. It just sets up an open disjoint union with an open complement. (This is exactly what is_closed tests.)

The purpose of this is to build a connection of manifolds to cell complexes and convex polyhedra: In a separate ticket, we will define embedded submanifolds of euclidean spaces that arise as interiors of polyhedra or relative interiors of their faces.

[gh-mjungmath]

That sounds extremely interesting!

Btw: Do you really mean cell complexes or rather simplicial complexes?

[mkoeppe]

Simplicial is too narrow - that would not be enough (without triangulating) to represent general convex polyhedra and their boundary structure. So need something slightly more general.

[mkoeppe]

New commits:

​331aa59

sage.manifolds.closure_topological_submanifold, TopologicalSubmanifold.closure: New

[mkoeppe]

Here's a beginning.

Currently, is there already a way to get the image of the embedding of the submanifold in the ambient manifold as a "subset"?

[gh-mjungmath]

Replying to mkoeppe:

New commits:

​331aa59

sage.manifolds.closure_topological_submanifold, TopologicalSubmanifold.closure: New

Looks already nice!

Replying to mkoeppe:

Simplicial is too narrow - that would not be enough (without triangulating) to represent general convex polyhedra and their boundary structure. So need something slightly more general.

Right, the cube is already a counter-example.

Replying to mkoeppe:

Here's a beginning.

Currently, is there already a way to get the image of the embedding of the submanifold in the ambient manifold as a "subset"?

No, I don't think so.

[egourgoulhon]

Replying to gh-mjungmath:

Replying to mkoeppe:

Here's a beginning.

Currently, is there already a way to get the image of the embedding of the submanifold in the ambient manifold as a "subset"?

No, I don't think so.

I confirm; more generally, there is no such functionality for continuous maps.

[gh-mjungmath]

Speaking of boundary; I think we should slowly but surely consider to introduce manifolds with boundary...

[egourgoulhon]

Replying to gh-mjungmath:

Speaking of boundary; I think we should slowly but surely consider to introduce manifolds with boundary...

Indeed. There is even some demand for it: ​https://ask.sagemath.org/question/56532/.

[mkoeppe]

Replying to egourgoulhon:

Replying to gh-mjungmath:

Replying to mkoeppe:

Currently, is there already a way to get the image of the embedding of the submanifold in the ambient manifold as a "subset"?

No, I don't think so.

I confirm; more generally, there is no such functionality for continuous maps.

OK, I have opened #31653 for this

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​0579188

Fixup

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. Last 10 new commits:

​26c7e56	src/sage/manifolds/continuous_map_image.py: Update doctests
​3e273bb	TopologicalSubmanifold.as_subset: New
​9726d36	Docstring work
​19762ae	ImageManifoldSubset: New parameter domain_subset, use it in ContinuousMap.image
​964f9f7	src/sage/manifolds/continuous_map.py: Update copyright
​e711215	src/sage/manifolds/continuous_map_image.py: Add tests
​0f3e36d	Link in documentation of sage.manifolds.continuous_map_image
​218117b	sage.manifolds.closure_topological_submanifold, TopologicalSubmanifold.closure: New
​3b3662e	Fixup
​0666fae	Change to ClosureOfManifoldSubset

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​426e087	ManifoldSubset.declare_{sub,super}set: New
​72294f0	ManifoldSubset.closure: New

[git]

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

​6772570	ManifoldSubset.union: Handle arbitrary unions
​edfea33	ManifoldSubset.union: Add example and plots
​fdd0aba	ManifoldSubset.declare_union: Handle arbitrary unions
​c2f484c	src/sage/manifolds/subset.py: import itertools
​a083569	ManifoldSubset._intersection_subset, _union_subset: Use declare_subset, declare_superset; fix doctest
​da91653	ManifoldSubset._union_subset: Fixup
​3e6a9cf	Merge #31764
​3786cee	ManifoldSubset.difference, complement: New
​decd831	Merge #31798
​0faec94	ManifoldSubset.is_closed, declare_closed: New

[mkoeppe]

I have put the new class in a new subpackage sage.manifolds.subsets (see ​https://trac.sagemath.org/ticket/30139#comment:33)

[egourgoulhon]

Looks nice, but there are several issues reported by the patchbot, including doctest failures.

[gh-mjungmath]

Looks like the failures are originated in #31798.

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​94edd68	ManifoldSubset.declare_superset: Fix documentation
​41826b4	ManifoldSubset.declare_{sub,super}set: Expand docstring
​1c14655	Merge #31763
​6370e88	ManifoldSubset.difference, complement: New
​73eceac	Merge #31798
​2028258	sage.manifolds.subsets.closure, TopologicalSubmanifold.closure: New
​38c9e96	ManifoldSubset.is_closed, declare_closed: New

[mkoeppe]

Redid the branch on top of only #31763, #31798

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​3ef5141	Merge #31763
​79a625b	ManifoldSubset._reduce_intersection_members: Add examples, raise TypeError if input is empty family
​8e70023	ManifoldSubset._reduce_union_members: Add examples
​ee4efc9	ManifoldSubset._{union,intersection}_subset: Do cache the result here; add examples
​e7f7a7d	Merge #31764
​9259a2c	Merge #31798

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​ac53984

src/sage/manifolds/subsets/__init__.py: New

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​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

[gh-mjungmath]

One doctest has failed. Is that related to this ticket?

[mkoeppe]

No, it's unrelated, see #31848

[egourgoulhon]

Looks nice!

There is a spurious line feed in the html documentation of ManifoldSubsetClosure, which can be fixed with

     - ``name`` -- (default: computed from the name of the subset)
-       string; name (symbol) given to the closure
+      string; name (symbol) given to the closure

Besides, there should be some EXAMPLES section in the main docstring of ManifoldSubsetClosure, and the latter should start with r""", instead of """, I think.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​b9909c0

ManifoldSubsetClosure: Add examples, fix docstring markup

[egourgoulhon]

I would not perform the import of ManifoldSubsetClosure in the EXAMPLES section, but rather construct the closure via the dedicated method closure(), since this is what the end user is supposed to do. In other words, I would rewrite the first part of the example as something like

sage: M = Manifold(2, 'R^2', structure='topological')
sage: c_cart.<x,y> = M.chart() # Cartesian coordinates on R^2
sage: D = M.open_subset('D', coord_def={c_cart: x^2+y^2<1}); D
Open subset D of the 2-dimensional topological manifold R^2
sage: cl_D = D.closure()
sage: cl_D
Topological closure cl_D of the Open subset D of the 2-dimensional
 topological manifold R^2
sage: latex(cl_D)
\mathop{\mathrm{cl}}(D)
sage: type(cl_D)
<class 'sage.manifolds.subsets.closure.ManifoldSubsetClosure_with_category'>
sage: cl_D.category()
Category of subobjects of sets

[egourgoulhon]

Also, in the docstring of ManifoldSubset.closure(), it would be nice to add an OUTPUT field as follows:

OUTPUT:

- an instance of :class:`~sage.manifolds.subsets.closure.ManifoldSubsetClosure`

This allows one to easily access to the documentation of class ManifoldSubsetClosure from that of closure(). For completeness, one may also add an INPUT field describing the arguments name and latex_name and their default values.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​16c6a72

ManifoldSubsetClosure, ManifoldSubset.closure: Improve documentation

[mkoeppe]

Thanks for the suggestions, done.

[egourgoulhon]

Thanks!

[mkoeppe]

Thank you!

[git]

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

​359dde1	Merge branch 't/31727/manifoldsubset__add_methods_subset_family__superset_family__equal_subset_family__deprecate_method_list_of_subsets' into t/31732/manifoldsubset__new_methods_declare_empty__declare_nonempty__is_empty__has_defined_points__open_cover_family
​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

[mkoeppe]

Trivial merge of updated dependency #31763