How is Every Prime z-Filter Contained in a Unique z-Ultrafilter in T_3 Space?

[chhan92]

Can you help me on this problem please?
I tried searching online, but I cannot find the proof:

In T_3 space (or regular and T_1 (any one-point set is closed)), show that every prime z-filter is contained in a unique z-ultrafilter. I feel so stupid because I spent lots of time and I cannot still do it.

 

[mathwonk]

try telling us clearly what the words mean.

 

[chhan92]

z-filter is the collection F of nonempty zero sets (f^{-1}(0) of continuous f:X -> I) such that
a) P_1, P_2 in F implies P_1 intersection P_2 in F
b) P_1 in F and a zero set P_2 containing P_1 implies P_2 in F.

A z-filter is prime if P_1 and P_2 belong to set of zero sets and P_1 union P_2 in F, then P_1 is in F or P_2 is in F.

An z-ultrafilter is a maximal z-filter.

As this exercise is from Willard, T_3 means that it is regular and T_1 (where all single point sets are closed)