Let $X$ be an affine algebraic subset of the $n$-dimensional affine space $A^n$ over a field $k$, and $A$ the affine line, both equipped with the Zariski topology. Is it true that a continuous function $f: X\rightarrow A$ is always the restriction of a polynomial in $k[x_1,...,x_n]$ to $X$?


closed as off-topic by Shaun, Namaste, Mohammad Riazi-Kermani, Chris Custer, Saad Apr 4 '18 at 1:05

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  • $\begingroup$ What kind of approaches (to similar problems) are you familiar with? What kind of answer are you looking for: Basic approach, hint, explanation, something else? Where did you encounter this problem? Have you tried anything? $\endgroup$ – Shaun Mar 28 '18 at 1:54
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    $\begingroup$ @Shaun I realize now that it's not true. We only have to take $X=A$ and any one-to-one function is continuous for the Zariski topology. Is that right? $\endgroup$ – Jiu Mar 28 '18 at 1:58
  • $\begingroup$ I don't know. ${}$ $\endgroup$ – Shaun Mar 28 '18 at 1:59

The topology on the affine line is the cofinite topology, so any bijection $\mathbb{A}^1\to\mathbb{A}^1$ is continuous. Certainly not all of these are given by polynomials, in general.


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