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For example, in a homework assignment I need to prove that a metric space $M$ is isometric to itself. If I say that $M$ maps to $M$ is that the same as saying that theres a function that takes elements from $M$ and I get a result in $M$? Or what is the "literal" meaning behind saying "map/maps/mapping".

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"Map" is not a precise word; if someone said that $M$ "maps to" $N$ it is safe to assume that there exists some function $f: M\to N$, but without more information it is hard to say anything more specific about $f$, although depending on the context it might mean that $f$ is a homeomorphism, diffeomorphism, isomorphism, etc.

In any case I would recommend writing "$f$ maps $M$ to $N$" over "$M$ maps to $N$," as the former has far more currency than the latter.

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It is essentially what you say. To be precise, you need to find a function $f$ that maps $M$ to $M$ and is isometric. There isn't really much more to it: a map is a functional relationship between two spaces (domain and range/codomain). It is used interchangeably as in "$f$ maps $M$ to $M$", "$f$ is a map from $M$ to $M$", or "$f$ is a function mapping $M$ to $M$".

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