What is a nontrivial ultranet?

  • Thread starter huyichen
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  • #1
huyichen
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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!
 

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  • #2
rasmhop
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First of all, I don't quite understand the meaning of a nontrivial ultranet, since the book itself does not give a precise definition.
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.

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
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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