  Denote by $\continuum=2^{\aleph_0}$ the cardinal of continuum. We construct
an intriguing family $(P_\alpha: \alpha\in\continuum)$ of prime $z$-ideals in
$\C_0(\reals)$ with the following properties:
  If $f\in P_{i_0}$ for some $i_0\in\continuum$, then $f\in P_i$ for all but
finitely many $i\in \continuum$;
  $\bigcap_{i\neq i_0} P_i \nsubset P_{i_0}$ for each $\i_0\in \continuum$.
  We also construct a well-ordered increasing chain, as well as a well-ordered
decreasing chain, of order type $\kappa$ of prime $z$-ideals in $\C_0(\reals)$
for any ordinal $\kappa$ of cardinality $\continuum$.
