Topological manifolds: basics

[egourgoulhon]

This is the implementation of topological manifolds over a topological field K resulting from the SageManifolds project. See the meta-ticket #18528 for an overview.
By topological manifold over a topological field K it is meant a second countable Hausdorff space M such that every point in M has a neighborhood homeomorphic to Kⁿ, with the same non-negative integer n for all points.

This tickets implements the following Python classes:

• ManifoldSubset: generic subset of a topological manifold (the open subsets being implemented by the subsclass TopologicalManifold)
  • TopologicalManifold: topological manifold over a topological field K
• ManifoldPoint: point in a topological manifold
• Chart: chart of a topological manifold
  • RealChart: chart of a topological manifold over the real field
• CoordChange: transition map between two charts of a topological manifold

as well as the singleton classesTopologicalStructure and RealTopologicalStructure.

TopologicalManifold is intended to serve as a base class for specific manifolds, like smooth manifolds (K=R) and complex manifolds (K=C). The follow-up ticket, implementing continuous functions to the base field, is #18640.

Documentation:
The reference manual is produced by
sage -docbuild reference/manifolds html
It can also be accessed online at http://sagemanifolds.obspm.fr/doc/18529/reference/manifolds/
More documentation (e.g. example worksheets) can be found here.

Depends on #18175

CC: @man74cio

Component: geometry

Keywords: topological manifolds

Author: Eric Gourgoulhon, Travis Scrimshaw

Branch/Commit: 00d265c

Reviewer: Travis Scrimshaw, Eric Gourgoulhon

Issue created by migration from https://trac.sagemath.org/ticket/18529

[egourgoulhon]

Description changed:

--- 
+++ 
@@ -1 +1,11 @@
-This is the implementation of topological manifolds over some field K resulting from the [SageManifolds project](http://sagemanifolds.obspm.fr/). See the meta-ticket #18528 for an overview.
+This is the implementation of topological manifolds over a field K resulting from the [SageManifolds project](http://sagemanifolds.obspm.fr/). See the meta-ticket #18528 for an overview.
+
+This tickets implements the following Python classes:
+
+- `TopManifold`: topological manifold over a field K
+- `TopManifoldPoint`: point in a topological manifold
+- `TopManifoldSubset`: generic subset of a topological manifold
+  - `TopManifoldOpenSubset`: open subset of a topological manifold
+- `Chart`: chart of a topological manifold
+  - `RealChart`: chart of a topological manifold over the real field
+- `CoordChange`: transition map between two charts of a topological manifold

[jhpalmieri]

comment:2

The phrase "manifold over a field K" sounds odd to me. Is it used in the literature? What if K is a finite field? It seems that if X is a finite discrete space, for every finite field F and for every non-negative integer n, then X is a manifold over F of dimension n: F and n play no role. (I'm assuming that finite fields have been given the discrete topology.)

I think you might say "topological manifold over a topological field K", since obviously the topology on K is critical. Or you could omit "over a field K", and mention in the documentation that users can specify a topological field (like \CC, rather than the default \RR) if they want.

[egourgoulhon]

Description changed:

--- 
+++ 
@@ -1,11 +1,14 @@
-This is the implementation of topological manifolds over a field K resulting from the [SageManifolds project](http://sagemanifolds.obspm.fr/). See the meta-ticket #18528 for an overview.
+This is the implementation of topological manifolds over a topological field K resulting from the [SageManifolds project](http://sagemanifolds.obspm.fr/). See the meta-ticket #18528 for an overview.
+By *topological manifold over a topological field K* it is meant a second countable Hausdorff space M such that every point in M has a neighborhood homeomorphic to K<sup>n</sup>, with the same non-negative integer n for all points. 
 
 This tickets implements the following Python classes:
 
-- `TopManifold`: topological manifold over a field K
+- `TopManifold`: topological manifold over a topological field K
 - `TopManifoldPoint`: point in a topological manifold
 - `TopManifoldSubset`: generic subset of a topological manifold
   - `TopManifoldOpenSubset`: open subset of a topological manifold
 - `Chart`: chart of a topological manifold
   - `RealChart`: chart of a topological manifold over the real field
 - `CoordChange`: transition map between two charts of a topological manifold
+
+`TopManifold` is intended to serve as a base class for specific manifolds, like smooth manifolds (K=**R**) and complex manifolds (K=**C**). 

[egourgoulhon]

comment:4

Replying to @jhpalmieri:

The phrase "manifold over a field K" sounds odd to me. Is it used in the literature?

Thanks for your comment. You are right: this is an abusive generalization of "manifold over R" and "manifold over C", which are used in the literature.

What if K is a finite field? It seems that if X is a finite discrete space, for every finite field F and for every non-negative integer n, then X is a manifold over F of dimension n: F and n play no role. (I'm assuming that finite fields have been given the discrete topology.)

I think you might say "topological manifold over a topological field K", since obviously the topology on K is critical. Or you could omit "over a field K", and mention in the documentation that users can specify a topological field (like \CC, rather than the default \RR) if they want.

Thanks for your suggestion; I've modified the ticket description accordingly. I've also added what is meant by "topological manifold over a topological field K".

PS: note that the code in the associated branch is still in a very crude draft state, but should be ready for review within a few days.

[sagetrac-git]

Changed commit from 89c063c to 5a5722b

[sagetrac-git]

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

5a5722b

Add doctests in classes Chart and RealChart

[sagetrac-git]

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

4f490af

Improve the documentation of coordinate charts

[sagetrac-git]

Changed commit from 5a5722b to 4f490af

[sagetrac-git]

Changed commit from 4f490af to d8df59f