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Lets start off with the definition of diffeomorphism from Wolfram MathWorld:

A diffeomorphism is a map between manifolds which is differentiable and has a differentiable inverse.

The issue is that I am learning about smooth manifolds, and in the books I've read, the map has to be smooth and have a smooth inverse. Also, the definition above doesn't say that it has to be bijective. However, the Encyclopedia of Math does.

So, are these definitions equivalent? If not, when are they equivalent?