Let $k$ be an [algebraically closed field](/page/Algebraically%20Closed%20Field), let $n\in \mathbb N$, and let $X\subseteq \mathbb A_k^n$ be a nonempty irreducible affine variety with vanishing ideal $I(X)\trianglelefteq k[x_1,\dots,x_n]$. For $p\in X$, write $\mathcal O_{X,p}$ for the local ring of $X$ at $p$, and say that $p$ is a smooth point of $X$ if $\mathcal O_{X,p}$ is a regular local ring. Define the smooth locus of $X$ by

\begin{align*} X_{\mathrm{sm}}:=\{p\in X: p \text{ is a smooth point of } X\}. \end{align*}

Then $X_{\mathrm{sm}}$ is a nonempty open subset of $X$ in the Zariski topology. In particular, $X_{\mathrm{sm}}$ is dense in $X$.