atlas

[meteor]

I have printed a notes about differential geometry, and it says:
-A Coo differentiable structure on a locally Euclidean, Hausdorff topological space M of dimension m is a collection of coordinate systems F
Then it specifies the conditions that F must satisfy, but I'm a little lazy and won't write it
Then it says:
-A C00 differentiable structure F which is maximal is called an atlas.
Then the text do not specify what it means by maximal. this is my doubt, what is a maximal C00 differentiable structure

[lethe]

Originally posted by meteor
I have printed a notes about differential geometry, and it says:
-A Coo differentiable structure on a locally Euclidean, Hausdorff topological space M of dimension m is a collection of coordinate systems F
Then it specifies the conditions that F must satisfy, but I'm a little lazy and won't write it
Then it says:
-A C00 differentiable structure F which is maximal is called an atlas.
Then the text do not specify what it means by maximal. this is my doubt, what is a maximal C00 differentiable structure
a set of charts satisfying those requirements that you alluded to is called maximal if any other set of charts which satisfies the conditions is a subset of this one.

i find it a little more comfortable to call any set of charts that satisfies the conditions an atlas. then the above sentence is a little easier to read:

an atlas is maximal if any other atlas on the space is a subset.

by Zorn's Lemma, any space with an atlas has a maximal atlas.

[Hurkyl]

In general, the term "maximal" means that there is nothing bigger than it. In many cases, though, you can prove that something maximal is bigger than everything else (such as in this case)