How to prove that SU(3) is compact

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I think for any map M: A->B where A and B are some smooth manifolds, it is enough for U<M to be a manifold that it is the inverse image of some fixed point p in B and that M is a surjection (or has rank equal to dimension of B). That is because you can compose M with the coordinate charts of B which are full rank :)
 

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