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So the question asks to find a topology on $\mathbb{R}$ so that all the polynomials are continuous functions from $\mathbb{R}$ to $\mathbb{R}$ but $\cos(x)$ is not continuous as a function from $\mathbb{R}$ to $\mathbb{R}$.

I looked at another similar question in stack exchange: A Topology such that the continuous functions are exactly the polynomials I think this is a special case of my question since it requires all other functions to not be continuous.

Other than that I thought maybe the solution could be based on how cosine is a periodic function unlike the polynomials, so maybe I should take out some infinite sets from the topology but that still wouldn't be enough since the topology would be closed under unions. So I am not even sure if there exists such topology.

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    $\begingroup$ How about the cofinite topology? $\endgroup$ – Lord Shark the Unknown Feb 25 at 19:32
  • $\begingroup$ Could you please explain how the in the cofinite topology the cosine function would not be continuous. $\endgroup$ – kkslblg Feb 25 at 19:51
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Let's put the cofinite topology on the real line, and suppose that the cosine function is continuous. Than the counterimage of $0$ should be closed; but it has infinitely many points, and in the cofinite topology a subset is closed iff it is finite.

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