Does a non-empty locally closed subset of a $k$-scheme of finite type always contain a closed point? Let $k$ be a field.
Let $X$ be a scheme of finite type over $k$.
We denote by $X_0$ the set of closed points of $X$.
Let $Y$ be a locally closed subset of $X$.
Is $Y \cap X_0$ non-empty whenever $Y$ is non-empty?

Motivation
See my comment to Avi Steiner's answer to this question.
Write $Y=Z \cap U$ with $Z \subseteq X$ closed and $U \subseteq X$ open. Now we may view $Y$ as an open subset of the closed subset $Z$ of $X$. We make $Z$ into a closed subscheme of $X$. Then it is again of finite type over $k$, and we have $Z_0 = X_0 \cap Z$. Thereby we reduce the problem to open subsets. But then it follows from a corollary of Hilbert's Nullstellensatz, stating that the closed points are dense.
As Martin Brandenburg's answer shows that we may suppose that $Y$ is a non-empty open subset of $X$.
Then $Y \cap X_0$ is non-empty because $X_0$ is dense in $X$ by the proposition of my answer to this question.