What Is a Nontrivial Ultranet and Why Is the Axiom of Choice Needed?

[huyichen]

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!

 

[rasmhop]

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 $\Lambda$ and fixed element $x \in X$, the map $P : \Lambda \to X$ defined by $P(\lambda)=x$ 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.