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.
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)