the notion of complete atlas











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The following picture comes from foundations of differential geometry volume I.enter image description here



According to the definition of complete atlas, would it be more precise to call it maximal atlas?



Secondly, the statement




Every atlas of $M$ compatible with $Gamma$ is contained in a unique atlas of $M$ compatible with $Gamma$.




can be proved routinely:



Let $mathscr A={mbox{all the atlases of }M mbox{ compatible with }Gammambox{ that contains }A}$, I guess it is a set.
For each chain $mathscr A_m$ in $mathscr A$, let $A_m=cup mathscr A_m$, then it is in $mathscr A$: Suppose $(U_1,phi_1)in A_1in mathscr A_m, (U_2,phi_2)in A_2in mathscr A_m$ and $A_1subset A_2$, then $phi_2circphi_1^{-1}:phi_1(U_1cap U_2)tophi_2(U_1cap U_2) $ is an element of $Gamma$ whenever $U_1cap U_2$ is nonempty, since $(U_1,phi_1)$ and $(U_2,phi_2)$ are both in $A_2$. Hence $A_m$ is an bound of the chain $mathscr A_m$. By Zorn's lemma, there is a maximal element in $mathscr A$.



Can someone verify for me if the proof is correct?






differential-geometry notation manifolds







share|cite|improve this question




















  • 4




    Are you asking something?
    – Will M.
    19 hours ago










  • Oh, I'm asking if it is more precise to call the complete atlas maximal atlas, and if my proof is right.
    – C.Ding
    19 hours ago






  • 2




    I knew it as "Saturated Atlas" but sure, maximal atlas is better. Your proof is correct but it is unnecessary to appeal to Zorn's lemma. Just take the union of all atlases compatible with $A$ to get the maximal atlas.
    – Will M.
    19 hours ago










  • @WillM. But I used the condition of the chain in my proof, how to prove without that?
    – C.Ding
    19 hours ago












  • @WillM How to prove $phi_2circphi_1^{-1}in Gamma$ without using $A_1subset A_2$?
    – C.Ding
    18 hours ago

















up vote
0
down vote

favorite












The following picture comes from foundations of differential geometry volume I.enter image description here



According to the definition of complete atlas, would it be more precise to call it maximal atlas?



Secondly, the statement




Every atlas of $M$ compatible with $Gamma$ is contained in a unique atlas of $M$ compatible with $Gamma$.




can be proved routinely:



Let $mathscr A={mbox{all the atlases of }M mbox{ compatible with }Gammambox{ that contains }A}$, I guess it is a set.
For each chain $mathscr A_m$ in $mathscr A$, let $A_m=cup mathscr A_m$, then it is in $mathscr A$: Suppose $(U_1,phi_1)in A_1in mathscr A_m, (U_2,phi_2)in A_2in mathscr A_m$ and $A_1subset A_2$, then $phi_2circphi_1^{-1}:phi_1(U_1cap U_2)tophi_2(U_1cap U_2) $ is an element of $Gamma$ whenever $U_1cap U_2$ is nonempty, since $(U_1,phi_1)$ and $(U_2,phi_2)$ are both in $A_2$. Hence $A_m$ is an bound of the chain $mathscr A_m$. By Zorn's lemma, there is a maximal element in $mathscr A$.



Can someone verify for me if the proof is correct?






differential-geometry notation manifolds







share|cite|improve this question




















  • 4




    Are you asking something?
    – Will M.
    19 hours ago










  • Oh, I'm asking if it is more precise to call the complete atlas maximal atlas, and if my proof is right.
    – C.Ding
    19 hours ago






  • 2




    I knew it as "Saturated Atlas" but sure, maximal atlas is better. Your proof is correct but it is unnecessary to appeal to Zorn's lemma. Just take the union of all atlases compatible with $A$ to get the maximal atlas.
    – Will M.
    19 hours ago










  • @WillM. But I used the condition of the chain in my proof, how to prove without that?
    – C.Ding
    19 hours ago












  • @WillM How to prove $phi_2circphi_1^{-1}in Gamma$ without using $A_1subset A_2$?
    – C.Ding
    18 hours ago















up vote
0
down vote

favorite









up vote
0
down vote

favorite











The following picture comes from foundations of differential geometry volume I.enter image description here



According to the definition of complete atlas, would it be more precise to call it maximal atlas?



Secondly, the statement




Every atlas of $M$ compatible with $Gamma$ is contained in a unique atlas of $M$ compatible with $Gamma$.




can be proved routinely:



Let $mathscr A={mbox{all the atlases of }M mbox{ compatible with }Gammambox{ that contains }A}$, I guess it is a set.
For each chain $mathscr A_m$ in $mathscr A$, let $A_m=cup mathscr A_m$, then it is in $mathscr A$: Suppose $(U_1,phi_1)in A_1in mathscr A_m, (U_2,phi_2)in A_2in mathscr A_m$ and $A_1subset A_2$, then $phi_2circphi_1^{-1}:phi_1(U_1cap U_2)tophi_2(U_1cap U_2) $ is an element of $Gamma$ whenever $U_1cap U_2$ is nonempty, since $(U_1,phi_1)$ and $(U_2,phi_2)$ are both in $A_2$. Hence $A_m$ is an bound of the chain $mathscr A_m$. By Zorn's lemma, there is a maximal element in $mathscr A$.



Can someone verify for me if the proof is correct?






differential-geometry notation manifolds







share|cite|improve this question















The following picture comes from foundations of differential geometry volume I.enter image description here



According to the definition of complete atlas, would it be more precise to call it maximal atlas?



Secondly, the statement




Every atlas of $M$ compatible with $Gamma$ is contained in a unique atlas of $M$ compatible with $Gamma$.




can be proved routinely:



Let $mathscr A={mbox{all the atlases of }M mbox{ compatible with }Gammambox{ that contains }A}$, I guess it is a set.
For each chain $mathscr A_m$ in $mathscr A$, let $A_m=cup mathscr A_m$, then it is in $mathscr A$: Suppose $(U_1,phi_1)in A_1in mathscr A_m, (U_2,phi_2)in A_2in mathscr A_m$ and $A_1subset A_2$, then $phi_2circphi_1^{-1}:phi_1(U_1cap U_2)tophi_2(U_1cap U_2) $ is an element of $Gamma$ whenever $U_1cap U_2$ is nonempty, since $(U_1,phi_1)$ and $(U_2,phi_2)$ are both in $A_2$. Hence $A_m$ is an bound of the chain $mathscr A_m$. By Zorn's lemma, there is a maximal element in $mathscr A$.



Can someone verify for me if the proof is correct?






differential-geometry notation manifolds




differential-geometry notation manifolds






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 17 hours ago









Brahadeesh

5,52941956




5,52941956










asked 19 hours ago









C.Ding

1,2311321




1,2311321








  • 4




    Are you asking something?
    – Will M.
    19 hours ago










  • Oh, I'm asking if it is more precise to call the complete atlas maximal atlas, and if my proof is right.
    – C.Ding
    19 hours ago






  • 2




    I knew it as "Saturated Atlas" but sure, maximal atlas is better. Your proof is correct but it is unnecessary to appeal to Zorn's lemma. Just take the union of all atlases compatible with $A$ to get the maximal atlas.
    – Will M.
    19 hours ago










  • @WillM. But I used the condition of the chain in my proof, how to prove without that?
    – C.Ding
    19 hours ago












  • @WillM How to prove $phi_2circphi_1^{-1}in Gamma$ without using $A_1subset A_2$?
    – C.Ding
    18 hours ago
















  • 4




    Are you asking something?
    – Will M.
    19 hours ago










  • Oh, I'm asking if it is more precise to call the complete atlas maximal atlas, and if my proof is right.
    – C.Ding
    19 hours ago






  • 2




    I knew it as "Saturated Atlas" but sure, maximal atlas is better. Your proof is correct but it is unnecessary to appeal to Zorn's lemma. Just take the union of all atlases compatible with $A$ to get the maximal atlas.
    – Will M.
    19 hours ago










  • @WillM. But I used the condition of the chain in my proof, how to prove without that?
    – C.Ding
    19 hours ago












  • @WillM How to prove $phi_2circphi_1^{-1}in Gamma$ without using $A_1subset A_2$?
    – C.Ding
    18 hours ago










4




4




Are you asking something?
– Will M.
19 hours ago




Are you asking something?
– Will M.
19 hours ago












Oh, I'm asking if it is more precise to call the complete atlas maximal atlas, and if my proof is right.
– C.Ding
19 hours ago




Oh, I'm asking if it is more precise to call the complete atlas maximal atlas, and if my proof is right.
– C.Ding
19 hours ago




2




2




I knew it as "Saturated Atlas" but sure, maximal atlas is better. Your proof is correct but it is unnecessary to appeal to Zorn's lemma. Just take the union of all atlases compatible with $A$ to get the maximal atlas.
– Will M.
19 hours ago




I knew it as "Saturated Atlas" but sure, maximal atlas is better. Your proof is correct but it is unnecessary to appeal to Zorn's lemma. Just take the union of all atlases compatible with $A$ to get the maximal atlas.
– Will M.
19 hours ago












@WillM. But I used the condition of the chain in my proof, how to prove without that?
– C.Ding
19 hours ago






@WillM. But I used the condition of the chain in my proof, how to prove without that?
– C.Ding
19 hours ago














@WillM How to prove $phi_2circphi_1^{-1}in Gamma$ without using $A_1subset A_2$?
– C.Ding
18 hours ago






@WillM How to prove $phi_2circphi_1^{-1}in Gamma$ without using $A_1subset A_2$?
– C.Ding
18 hours ago

















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