EDIT: I apologize for the confusion by which I originally framed this question for normal spaces, where it has an uninteresting answer (thanks to those who pointed this out). Hope I've got it right now.

Given a (Hausdorff/regular/Tychonoff) space $X$, attach to each ordinal a family of sets, as follows:

1) $H(0) = \{X\}$;

2) $H(\kappa + 1) = $ the family of all $Z$-sets (zero sets of continuous functions) of sets in $\bigcup_{\lambda \leq \kappa} H(\lambda) $;

3) for non-zero limit ordinal $\kappa$, $H(\kappa) = $ the family of all (nested? does it make a difference?) intersections of sets in $\bigcup_{\lambda < \kappa} H(\lambda) $

Associate to $X$, $\nu=\nu(X)$, the smallest ordinal such that $H(\nu)=H(\lambda)$ for all $\lambda > \nu$, i.e., the ordinal where the hierarchy collapses.

What ordinals occur as $\nu(X)$ for some (Hausdorff/regular/Tychonoff) space $X$? How does one build examples for those ordinal that do so occur?