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:

• TopManifold: topological manifold over a topological field K
• TopManifoldPoint: point in a topological manifold
• TopManifoldSubset: generic 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).

[jhpalmieri]

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]

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.

[git]

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

​5a5722b

Add doctests in classes Chart and RealChart

[git]

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

​4f490af

Improve the documentation of coordinate charts

[git]

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

​d8df59f

Improve the documentation of TopManifold

[git]

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

​fb96562

Open subsets of topological manifolds are now fully considered as topological manifolds.