Let $X$ and $Y$ be Noetherian schemes over $\mathbb{C}$ and suppose there is a $1-1$-correspondence between closed points of $X$ and that of $Y$. Does this imply that the dimension of $X$ is the same as the dimension of $Y$?



Topological properties of schemes
No. For instance, $X=\textrm{Spec }\mathbb C[[t]]$ and $Y=\textrm{Spec }\mathbb C$ both have one closed point, but $\dim X=1$ and $\dim Y=0$.


