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Again, I am reading this. I am finding it a bit difficult to understand the definition of n-dimensional smooth manifold.

Now,

$\{U_a; x^1_a, x^2_a, ..., x^n_a\}$ ----(1)

Is the thing (1) a set? (I think it is not).

Is it a tuple?

Also, is $U_a$ a set or a set of sets? What is significance of the subscript $a$?

I would like it very much if someone explains the definition easier to understand. With good examples.

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This is there in the notes. $\{ U_a \}$ is an open cover of $M$. This is a set where each element $U_a$ is an open subset of $M$. The $a$ is used as an index for this set; so it would help to have put $a\in A$, say, where appropriate.

Each $x_a^i$ is a coordinate function on the open set $U_a$ (for given $a$).

The notes do give an example of the unit sphere which has an open cover consisting of two open subsets.

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    Strictly speaking it is$a$pair. One element is the subset $U_a$ and the other is a list of functions on $U_a$.2010-12-31
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A n-dim manifold is local homeomorphism like a n-dim Euclidean Space.In the smooth situation,you can put the open set(just a common part) of R^n on the any part of the mainfold with the diffeomorphism,it keeps the full rank. sphere is a good and basic example, you can white down the diffeomorphic mapping,but for many other figures that is complicated.