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lethe
#2
Mar14-04, 03:17 PM
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P: 657
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.