I some how could not find the definition of maximal atlas on a manifold.
What I see is that an atlas is said to be maximal atlas if it is not contained in any other atlas.
What does this containment actually mean?
Let $\mathcal{A}$ be an atlas and $\mathcal{B}$ be another atlas. When do we say that $\mathcal{A}$ is contained in $\mathcal{B}$?
I was not able to find definition of this.
Another confusion is about union of atlases. Let $\mathcal{A}$ and $\mathcal{B}$ be two atlases. What do we mean by union of atlases? Is it just the union $\{(U,\phi)_{\phi\in \mathcal{A}},(V,\psi)_{\psi\in \mathcal{B}}\}$?
It may happen that this union is not an atlas i.e., there can be two charts $\phi_\mathcal{A}$ and $\psi_{\mathcal{B}}$such that $\phi_{\mathcal{A}}$ and $\psi_{\mathcal{B}}$ are not compatible.
By maximal atlas do I mean an atlas $\mathcal{A}$ such that for any other atlas $\mathcal{B}$, the union as above is not an atlas?
Any reference for the definition is welcome.
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$\begingroup$"Contain" and "union" here literally mean just that. An atlas $\mathcal{A}$ is a set of charts $(U,\phi)$, and $\mathcal{A}$ is contained in $\mathcal{B}$ if $\mathcal{A}\subseteq\mathcal{B}$: that is, if every chart which is an element of $\mathcal{A}$ is also an element of $\mathcal{B}$. The union of two atlases is just the set $\mathcal{A}\cup\mathcal{B}$, which as you observe may not be an atlas.
An atlas $\mathcal{A}$ is called maximal if there does not exist any atlas $\mathcal{B}$ such that $\mathcal{A}\subset\mathcal{B}$ (with a strict inclusion). This is equivalent to saying that if $\mathcal{B}$ is an atlas such that $\mathcal{A}\cup\mathcal{B}$ is an atlas, then $\mathcal{B}\subseteq\mathcal{A}$.
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