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- Thread starter huyichen
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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.First of all, I don't quite understand the meaning of a nontrivial ultranet, since the book itself does not give a precise definition.

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.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

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