# What is a nontrivial ultranet?

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!

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.