Equivalence of C^infinity atlases

Equivalence of C^\infty atlases is an equivalence relation. Each C^\infty atlas on M is equivalent to a unique maximal C^\infty atlas on M.

Proof:

Reflexive: If A is a C^\infty atlas, then A\cup A=A is also a C^\infty atlas.

Symmetry: Let A and B be two C^\infty atlases such that A\cup B is also a C^\infty atlas. Then certainly B\cup A is also a C^\infty atlas.

Transitivity: Let A, B, C be C^\infty atlases, such that A\cup B and B\cup C are both C^\infty atlases.

Notation:
\begin{aligned}  A&=\{(U_\alpha,\varphi_\alpha)\}\\  B&=\{(V_\beta, \psi_\beta)\}\\  C&=\{(W_\gamma, f_\gamma)\}.  \end{aligned}

Then \displaystyle \varphi_\alpha\circ f_\gamma^{-1}=\varphi_\alpha\circ\psi_\beta^{-1}\circ\psi_\beta\circ f_\gamma^{-1}: f_\gamma(U_\alpha\cap W_\gamma)\to\varphi_\alpha(U_\alpha\cap W_\gamma) is a diffeomorphism since both \varphi_\alpha\circ\psi_\beta^{-1} and \psi_\beta\circ f_\gamma^{-1} are diffeomorphisms due to A\cup B and B\cup C being C^\infty atlases. Also, M=\bigcup U_\alpha, M=\bigcup W_\gamma implies M=(\bigcup U_\alpha)\cup(\bigcup W_\gamma) so A\cup C is also a C^\infty atlas.

Let A be a C^\infty atlas on M. Define B to be the union of all C^\infty atlases equivalent to A. Then B\sim A. If B'\sim A, then B'\subseteq B, so that B is the unique maximal C^\infty atlas equivalent to A.

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