Uncountable families of prime z-ideals in C_0(R)

Abstract

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 $\continuum$.