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Frustrated by this one exercise

  1. Dec 11, 2011 #1
    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.
  2. jcsd
  3. Dec 12, 2011 #2


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    try telling us clearly what the words mean.
  4. Dec 12, 2011 #3
    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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