I really couldn't figure out the answer to the following question: Let $X$ be a scheme of finite type over a field $k$ and let $K$ be an extension field of $k$. Let $X_K := K \times_k X$ be the base extension and let $p: X_K \rightarrow X$ be the projection. Is it true that $p$ maps closed points to closed points?
This should be true if $K$ is of finite type over $k$ (so not necessarily finite!). Proof: $X$ is Jacobson, the projection $p$ is of finite type and now it follows for example from the Stacks Project Lemma 27.17.8 (see http://stacks.math.columbia.edu/tag/01TB) that $p$ maps closed points to closed points. But I would like to have the same for any $K$. Is this true?