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

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SUMMARY

In T_3 space, every prime z-filter is contained in a unique z-ultrafilter. A z-filter is defined as a collection of nonempty zero sets that satisfies specific intersection and containment properties. The uniqueness of the z-ultrafilter arises from its maximality, ensuring that no additional elements can be added without violating the filter properties. This conclusion is established through the definitions and properties of z-filters and z-ultrafilters as outlined in Willard's framework.

PREREQUISITES
  • Understanding of T_3 space and its properties (regular and T_1).
  • Familiarity with the concepts of z-filters and z-ultrafilters.
  • Knowledge of continuous functions and their zero sets.
  • Basic set theory, particularly regarding intersections and unions of sets.
NEXT STEPS
  • Study the properties of T_3 spaces in depth, focusing on regularity and closed sets.
  • Explore the concept of ultrafilters and their applications in topology.
  • Investigate the relationship between filters and ideals in set theory.
  • Review examples of z-filters and z-ultrafilters to solidify understanding of their properties.
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Mathematicians, particularly those specializing in topology, students studying advanced set theory, and anyone interested in the properties of filters and ultrafilters in mathematical analysis.

chhan92
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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.
 
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try telling us clearly what the words mean.
 
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)
 

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