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**slevvio** Let $\displaystyle \mathcal{Z} $ be the family of sets $\displaystyle U $ in $\displaystyle \mathbb{R} $ such that $\displaystyle U $ is empy or $\displaystyle R \setminus U $ is finite.

Show that $\displaystyle \mathcal{Z} $ is a topology on $\displaystyle \mathbb{R} $. By considering the inverse images of closed sets, show that every polynomial function $\displaystyle f: \mathbb{R} \rightarrow \mathbb{R} $ is $\displaystyle (\mathcal{Z},\mathcal{Z}) $-continuous.

Hey everyone, I was wondering if anyone could assist me with the 2nd part. If $\displaystyle A $ is closed in $\displaystyle \mathbb{R} $ then $\displaystyle \mathbb{R} \setminus A $ is the empty set, or A is finite, i.e. A is $\displaystyle \mathbb{R} $ or finite.

I don't understand though how $\displaystyle f^{-1} (A) $ is closed and I would really appreciate some advice here, thank you so much. This would show continuity of the polynomial functions.