What's a smooth neighborhood? Why can it result from a transition map?
Particularly I found this in a proof for the maximal atlas being smooth. Here one constructs a composition of two transition maps, but then it's said that the composition is "a smooth neighborhood of $x \in \mathbb{R}^n$ (which was shown earlier)".
But I wonder, how does this definition make sense.
I've thought neighborhoods are sets, but smoothness is a property of functions.
So how can one have smooth neighborhoods?
Maybe it tries to say that the image is a neighborhood of $x$, but the structure in hand is also a smooth structure.