# Topology on the real numbers such that all polynomials are continuous but cosine is not continuous

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.

• How about the cofinite topology? – Lord Shark the Unknown Feb 25 at 19:32
• Could you please explain how the in the cofinite topology the cosine function would not be continuous. – kkslblg Feb 25 at 19:51

## 1 Answer

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.