# Open subset of irreducible affine curve is an affine curve

How to show that every Zariski open subset of an irreducible affine curve is an affine curve? How is dimension defined for open subsets (without going to its coordinate rings)?

I know the definition of dimension for closed sets (which is the maximal chain of closed irreducible subsets).

• Regard the open subset as a topological space by using the subspace topology, then apply the usual definition of dimension. – Zhen Lin Apr 15 '13 at 18:53

To show that every Zariski open subset $U$ of an irreducible affine curve $C\subseteq \mathbb{A}^n$ is affine:
1) Show that there are only finitely many points $\{ \mathfrak{m}_P\}_{P\in C\backslash U}$ in $C \backslash U$.
2) Consider the localization $k[x_1,\cdots,x_n]_{\{\mathfrak{m}_P\}}/I_C$.