On Thu, 15 Nov 2012, David C. Ullrich wrote: > On Thu, 15 Nov 2012 01:28:04 -0800, William Elliot <firstname.lastname@example.org> > wrote: > > >What's an example of an universal or ultranet? > > > >Wikipedia claims that if n:D -> X is an ultranet into X and > >f:X -> Y, then the composition f.n:D -> Y is an ultranet into Y. > > > >I dispute the claim > > Did you spend more than a second thinking about it?
I did find an ultra net.
> The proof is completely and utterly mind-bogglingly > trvial. > > It relies on a few deep observations. Say f : X -> Y > and A is a subset of Y. Say x is an element of x. > > (i) Exactly one of the following holds: f(x) is in A, > f(x) is not in A. > > (ii) f(x) is not in A if and only if f(x) is in Y \ A.
Wow, that shows a function of a constant ultranet is an ultranet.
Here's what I was missing, to start with B below instead of A.
If B subset Y, then A = f^-1(B) subset X: net n eventually in A or eventually in X\A, net f.n eventuall in ff^-1(B) subset B or eventually in f(X\f^-1(B)) = ff^-1(Y\B) subset Y\B. QED.