Nonreduced double point ($\mathrm{Spec}k[ϵ]/({ϵ}^2)$)

The spectrum of the dual number ring $k[ϵ]/({ϵ}^2)$ is a nonreduced scheme of length two supported at one point. Its nonzero nilpotent $ϵ$ retains first-order information that the underlying singleton topological space forgets. It differs from the disjoint union of two reduced $k$-points.