Quick Smooth Manifold Question


MHF Hall of Honor
Nov 2009
Berkeley, California
This may be stupid question, but why is it that given a topological manifold \(\displaystyle X\) and some atlas \(\displaystyle \mathfrak{A}\) then their exists a unique \(\displaystyle C^{\infty}\) structure \(\displaystyle \mathfrak{A}^*\) on \(\displaystyle X\) which contains \(\displaystyle \mathfrak{A}\)?

I can see why their exists some \(\displaystyle C^{\infty}\) structure. Just define \(\displaystyle \Omega\) to be the set of all atlases on \(\displaystyle X\) containing \(\displaystyle \mathfrak{A}\), 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 \(\displaystyle \mathfrak{M},\mathfrak{N}\) would need to be comparable and thus equal?

Any help would be appreciated!