Could any one explain me in detail how to prove the following statements in rigorous way?
Two complex atlases are equivalent iff their union is also a complex atlas. An easy Zorn's lemma argument will show that every complex atlas is contained in a unique maximal complex atlas. Moreover, two atlases are equivalent iff they are contained in the same maximal complex atlas.
How to show that any atlas on $X$ determines a unique complex structure?
Thank you for your help.