If I'm going to take this renewed interest in trying to understand
differential geometry more seriously, I might as well read more than one
source on it. So, I'll start a sequence of posts on this book as well.

Just as Merkulov, Lee starts with a short definition of what a
topological manifold is, including definitions for the terms needed for
the treatment.

## Definition

An n-dimensional topological manifold is a second countable Haussdorff
space of local Euclidean dimension n.

Next, Lee goes on to define coordinate charts. I won't repeat the
treatment, since he doesn't really bring anything Merkulov hasn't talked
about in some manner or other. Atlases, smooth structures and
equivalence of atlases also merits some treatment.

The first really new thing I find is the Lemma 1.4. Lee points out that
for a topological manifold M, every smooth atlas is contained in a
unique maximal smooth atlas - this is probably a rather straightforward
application of Zorn's lemma, but Lee gives an explicit construction of
the unique maximal smooth atlas as theatlas of all charts that are
smoothly compatible with every chart in the original atlas. There are
some technicalities to be checked to verify that this is an atlas and
that it is maximal; but it all boils down to "because it's compatible
anyway".

Furthermore, for a second part he affirms that two smooth atlases
determine the same maximal smooth atlas iff their union is a smooth
atlas. The proof of this is left to the reader; but what would my
blogposts be if not the reader doing the things he's supposed to do?
So here goes.

Suppose two smooth atlases determine the same maximal smooth atlas.
Say [tex]\mathcal A[/tex] and [tex]\mathcal B[/tex]. We want to show
that then their union is a smooth atlas. Obviously their union is an
atlas, so we need only verify smoothness. So take some pair of charts
[tex](U_{\mathcal A},\phi_{\mathcal A})[/tex] and
[tex](U_{\mathcal B},\phi_{\mathcal B})[/tex]. We need to show
that on the respective images of [tex]U_{\mathcal A}\cap
U_{\mathcal B}[/tex] the functions [tex]\phi_{\mathcal
A}\circ\phi_{\mathcal B}^{-1}[/tex] and [tex]\phi_{\mathcal
A}\circ\phi_{\mathcal B}^{-1}[/tex] are smooth. So we pick some
arbritrary point [tex]x_0\in\phi_{\mathcal A}(U_{\mathcal
A}\cap U_{\mathcal B})[/tex] and want to show smoothness at this
point.

But we can use the maximal smooth atlas [tex]\mathcal C[/tex] now.
We pick a chart [tex](U_{\mathcal C},\phi_{\mathcal C})[/tex]
such that [tex]x_0\in U_{\mathcal C}[/tex]. Since this chart is
from the maximal smooth atlas, it is compatible with both
[tex]\mathcal A and \mathcal B[/tex]. More precisely, this means
that [tex]\phi_{\mathcal C}\circ\phi_{\mathcal A}^{-1}[/tex] is
smooth. So is also [tex]\phi_{\mathcal B}\circ\phi_{\mathcal
C}^{-1}[/tex]. So if we compose these two functions, we get a new
function. That takes points in [tex]\phi_{\mathcal A}(U_{\mathcal
A}\cap U_{\mathcal B})[/tex] to points in [tex]\phi_{\mathcal
B}(U_{\mathcal A}\cap U_{\mathcal B})[/tex]. And this function
is, as it is the composition of two smooth functions, also smooth.

For a different point we may have to pick a different chart from
[tex]\mathcal C[/tex], but since all charts in [tex]\mathcal C[/tex]
are smoothly compatible with all charts in both [tex]\mathcal A[/tex]
and [tex]\mathcal B[/tex], this won't really change much of the
argument. And thus we know that if both determine the same maximal
atlas then their union is a smooth atlas.

Suppose now that the two atlases [tex]\mathcal A[/tex] and
[tex]\mathcal B[/tex] have a smooth atlas as their union. That means
more specifically that all the charts in [tex]\mathcal B[/tex] are
smoothly compatible with all charts in [tex]\mathcal A[/tex], and thus
that [tex]\mathcal B[/tex] is contained in the maximal smooth atlas
containing [tex]\mathcal A[/tex] and vice versa. But since the maximal
smooth atlas containing a specific atlas was unique, they both have the
same maximal smooth atlas.

And this illuminates why the interesting condition Merkulov gave was
precisely that they have a smooth union.

Note that any manifold that can be covered by a single chart thus has a
smooth structure determined in its entirety by that chart.

And at this point - page 11 - Lee does something that I probably will
end up hating him for. He introduces the Einstein Summation Convention.
It's bad. It's ugly. And it stands for most of the things that led to my
flunking differential geometry in the first place. Boooo!

Another few pages yields an idiotic definition and the introduction of
diffeomorphisms without telling the reader what they are. I'll stop
reading this thing now.