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What is a nontrivial ultranet?

  1. Mar 21, 2010 #1
    As I read through Willard's General Topology, the problem 12 D-5 states that "the net based on a free ultrafilter is a nontrivial ultranet. Hence, assuming the axiom of choice, there are nontrivial ultranets." First of all, I don't quite understand the meaning of a nontrivial ultranet, since the book itself does not give a precise definition. Second of all, if "the net based on a free ultrafilter is a nontrivial ultranet" is already given, why do we still need axiom of choice to prove the existence of nontrivial ultranet?

    Help needed!
     
  2. jcsd
  3. Mar 24, 2010 #2
    The definition of an ultranet is given as definition 11.10. Just before 11.11 the author notes that for any directed set [itex]\Lambda[/itex] and fixed element [itex]x \in X[/itex], the map [itex]P : \Lambda \to X[/itex] defined by [itex]P(\lambda)=x[/itex] is an ultranet. Such ultranets are called trivial ultranets. Thus a non-trivial ultranet is an ultranet that is not a trivial ultranet.

    To prove the existence of a free ultrafilter Willard uses the axiom of choice. See the proof of theorem 12.12 and the remarks about the axiom of choice after it.
     
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