On countably generated $z$-ideals of $C(X)$ for first countable spaces

Abstract:

In paper [L], a question asked in [D] has been answered first by proving that: if $X$ is normal and first countable, then every countably generated $z$-ideal of $C(X)$ is pure; then, by giving an example of a nonpure countably generated $z$-ideal of $C(X)$ in a $\sigma$-compact (hence normal) but not first countable space $X$. In this paper a class $\mathcal {C}$ of topological spaces $X$ whose $C(X)$ has a nonpure countably generated $z$-ideal is constructed; it is proved that $\mathcal {C}$ contains a space $X$ which is first countable. So it is proved that in the proposition above the hypotheses "normal" and "first countable" are both essential. Finally in $\S 4$ I prove, as announced in [L], that if $X$ is a locally compact normal space, then every countably generated $z$-ideal of $C(X)$ is pure. For the terminology and notations see [GJ], [D], [L].