  Given a complete local (Noetherian) ring $T$, we find necessary and
sufficient conditions on $T$ such that there exists a local domain $A$ with
$|A| < |T|$ and $\widehat{A} = T$, where $\widehat{A}$ denotes the completion
of $A$ with respect to its maximal ideal. We then find necessary and sufficient
conditions on $T$ such that there exists a domain $A$ with $\widehat{A} = T$
and $|\mbox{Spec}(A)| < |\mbox{Spec}(T)|$. Finally, we use "partial
completions" to create local rings $A$ with $\widehat{A} = T$ such that
$\mbox{Spec}(A)$ has varying cardinality in different varieties.
