Lemma 53.19.13. Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme of dimension $1$. Let $Y \to X$ be an étale morphism. Let $y \in Y$ be a point with image $x \in X$. The following are equivalent
$x$ is a closed point of $X$ and a node, and
$y$ is a closed point of $Y$ and a node.