Subset of a scheme induced (ideal-theoretically) by a collection of given points

Question

Suppose we have an affine scheme $X=\operatorname{Spec}A$ and a collection $Y$ of certain points (corresponding to ideals of $A$). Is there a standard terminology to refer to the subset $X_Y$ of $X$ that consists of all the prime ideals of $X$ each of which lies inside some ideal in $Y$? For example if $Y$ consists of only one point $P$ then $X_Y$ is just "the local scheme of $X$ at $P$", usually denoted $X_P$. What if $Y$ has more than one points? Is there a standard name for $X_Y$? One option is "the local scheme of $X$ at $Y$", except that this might not be that "local" if $Y$ has many points.