Let $X$ be a locally noetherian scheme. Let $x \in X$ be such that $\{x\}$ is locally closed (i.e. is of the form $U \cap F$ where $U$ is open and $F$ closed). I want to show $\bar{ \{x\} }$ has its topological dimension $\leq 1$.
I tried to simplified the problem into a commutative algebra statement here Length of maximal ideal in noetherian ring but I get my self confused and I released it wasn't enough.
Do you have ideas?