  We study infinite intersections of open subschemes and the corresponding
intersection of Hilbert schemes. If $\{U_i\}$ is the collection of open
subschemes of a variety $X$ containing a fixed point $P$, then we show that the
Hilbert functor of flat and finite families on the spectrum of the local ring
of $P$ is given by the intersection of ${\Cal H}_i$, where ${\Cal H}_i$ is the
Hilbert functor of flat and finite families on $U_i$. In particular we show
that the Hilbert functor of flat and finite families on the spectrum of the
local ring of $P$ is representable by a scheme.
