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.

Depends on #31763 Depends on #31798

CC: @egourgoulhon @tscrim @yuan-zhou @mjungmath

Component: manifolds

Author: Matthias Koeppe

Branch/Commit: 9abc617

Reviewer: Eric Gourgoulhon

Issue created by migration from https://trac.sagemath.org/ticket/31644

[mjungmath]

comment:1

That sounds extremely interesting!

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

[mkoeppe]

comment:2

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]

Branch: u/mkoeppe/topological_closure_of_embedded_submanifolds

[mkoeppe]

Author: Matthias Koeppe, ...

[mkoeppe]

Commit: 331aa59

[mkoeppe]

New commits:

331aa59

sage.manifolds.closure_topological_submanifold, TopologicalSubmanifold.closure: New

[mkoeppe]

comment:5

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"?

[mjungmath]

comment:6

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]

comment:7

Replying to @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.

[mjungmath]

comment:8

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

[egourgoulhon]

comment:9

Replying to @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]

Dependencies: #31653

[mkoeppe]

comment:11

Replying to @egourgoulhon:

Replying to @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

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 331aa59 to 0579188

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

0579188

Fixup

[mkoeppe]

Work Issues: redo on top of #31653

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 0579188 to 0666fae

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

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

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 0666fae to 72294f0

[mkoeppe]

Changed author from Matthias Koeppe, ... to Matthias Koeppe

[mkoeppe]

Description changed:

--- 
+++ 
@@ -1,5 +1,6 @@
-We define a class for objects in the category of topological spaces that represents the topological closure of an embedded submanifold. 
-Embedded submanifolds will provide a method `closure` to construct it. As far as this ticket is concerned, this object is abstract: it only knows what object it is the closure of and the ambient space.
+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.

 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.

[mkoeppe]

Changed work issues from redo on top of #31653 to none

[mkoeppe]

Changed dependencies from #31653 to #31653, #31763

[mkoeppe]

Description changed:

--- 
+++ 
@@ -1,6 +1,6 @@
 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.
+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.

 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.

[mkoeppe]

Changed dependencies from #31653, #31763 to #31653, #31763, #31798

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 72294f0 to 0faec94

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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]

Description changed:

--- 
+++ 
@@ -2,5 +2,7 @@

 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.

[mkoeppe]

comment:25

I have put the new class in a new subpackage sage.manifolds.subsets (see #30139 comment:33)

[egourgoulhon]

comment:26

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

[mjungmath]

comment:27

Looks like the failures are originated in #31798.

[mkoeppe]

Changed dependencies from #31653, #31763, #31798 to #31763, #31798

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 0faec94 to 38c9e96

[mkoeppe]

comment:30

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

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 38c9e96 to 9259a2c

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

ac53984

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

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 9259a2c to ac53984

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from ac53984 to 60505b0

[mjungmath]

comment:35

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

[mkoeppe]

comment:36

No, it's unrelated, see #31848

[egourgoulhon]

comment:37

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.

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

b9909c0

ManifoldSubsetClosure: Add examples, fix docstring markup

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 60505b0 to b9909c0

[egourgoulhon]

comment:39

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]

comment:40

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.

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from b9909c0 to 16c6a72