What the underlying fibre set forgets

By Points and topology of a fibre, the underlying space of $X_s$ is the ordinary inverse image of $s$ with its subspace topology. Its local rings, however, are the quotients in Stalks of the scheme-theoretic fibre, not a structure determined just by that set. Nilpotents and the residue fields remain mathematical data. For example $k[ϵ]\mathrm{/}({ϵ}^2)$ has one prime and residue field $k$, yet $ϵ$ is a nonzero nilpotent, unlike the one-point reduced ring $k$.

For a general product $X{\times }_{S}Y$, even the underlying set requires more than a compatible pair $(x,y)$. By Points of a fibre product via residue-field tensors, the missing datum is a prime of $\kappa (x){\otimes }_{\kappa (s)}\kappa (y)$. Thus ordinary fibre topology and general fibre-product points must be kept distinct.