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Can you find a topology on the set of real numbers $\mathbb{R}$ so that all polynomials $p(x)=a_nx^n+...a_1x+a_0$ considered as functions from $\mathbb{R}$ to $\mathbb{R}$ are continuous functions but the cosine function $\cos \left(x\right)$ is not a continuous function from $\mathbb{R}$ to $\mathbb{R}$.

I tried cofinite topology and Zariski Topology but I didn't succesful. Can you help?

seth
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    I doubt it. Please [edit] the question to tell us why you think this might or might not be true, and where the question comes from. – Ethan Bolker Dec 27 '20 at 15:18

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Take the cofinite topology, in which the closed sets are $\mathbb{R}$ and the finite sets. If $F$ is finite, and $f$ is a polynomial, then $f^{-1}(F)$ is finite. Furthermore, $f^{-1}(\mathbb{R}) = \mathbb{R}$, so $f$ is continous. However, $\cos^{-1}(0)$ is infinite, but not equal to $\mathbb{R}$. Thus $\cos$ is not continuous.

J.-E. Pin
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