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Nicely done!

Initially I recoiled at the thought of the stiffness of the CD, but of course your absolutely right, at least for 2d manifolds.

Including the hole at the center of the CD?

there can be no hole on the cd shaped object, technically disc is the word to use but the point of the sentence is to provide a vivid picture

> you can put a cd shaped object on

You're thinking of open sets.

In particular, consider two intersecting planes. You can put all the discs you like on that surface, but it's not a manifold because on the line of intersection it's not locally R2.

The position of the points on the surface can be changed without worrying about the topology and its the topology we are worried about