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

[sagetrac-git]

Changed commit from 331aa59 to 0579188

[sagetrac-git]

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

0579188

Fixup