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Determine the image of the regular map $f: A^2 \to A^2$, $f(x,y)=(x,xy)$ and describe it from the point of view of topology. Would the image of f be $A^2$, because every point of $A^2$ is still in the image of f?

My topology-knowledge is very weak. So how would one go about answering the 2nd question?

Note: $A^2$ refers to two-dim affine space (the field is not given, so I am assuming that it is arbitrary).

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  • $\begingroup$ $A^2$ can mean a number of things. Is it the Euclidean plane here? $\endgroup$ Apr 5 '14 at 10:56
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    $\begingroup$ I am reasonably sure he means the affine plane over a field. $\endgroup$ Apr 5 '14 at 11:00
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The image is not $A^2$ because the point $(0,1)$ is not in the image. Indeed, the image is $(A^2\setminus Z(x))\cup\{ (0,0) \}$. When $x$ is nonzero, then you can find a preimage of $xy$. I am not sure what is ment by a topological description, but you can see that this is a constructible set.

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