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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.


$\{U_a; x^1_a, x^2_a, ..., x^n_a\}$ ----(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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up vote 4 down vote accepted

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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@BWW $U_a$ is open cover or union of all $\{U_a\}$ is open cover? Because point (a) in the definition says " $U_a$ is an open cover". – Pratik Deoghare Dec 31 '10 at 10:54
It says "The $U_a$" this means "The set of all $U_a$". So $\{ U_a\}$ is an open cover. – BWW Dec 31 '10 at 11:14
Well that answers one of the questions :)Thanks! But one remains what is this $\{U_a; x^1_a, x^2_a, ..., x^n_a\}$ object? See there is semicolon after $U_a$ and commas after each $x^r_a$. – Pratik Deoghare Dec 31 '10 at 11:32
Choose an $a$. Then $U_a$ is an open subset of $M$ and each $x_a^i$ (for $1\le i\le n$) is a function $U_a\rightarrow \mathbb{R}$ called a coordinate function. – BWW Dec 31 '10 at 11:53
Strictly speaking it is a pair. One element is the subset $U_a$ and the other is a list of functions on $U_a$. – BWW Dec 31 '10 at 11:55

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.

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