This may be stupid question, but why is it that given a topological manifold and some atlas then their exists a unique structure on which contains ?
I can see why their exists some structure. Just define to be the set of all atlases on containing , order it in the natural way and apply Zorn's lemma. But, why is it unique? Is it because the way one constructs the ordering any two maximal atlases would need to be comparable and thus equal?
Any help would be appreciated!