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Topic: Ultranet
Replies: 2   Last Post: Nov 16, 2012 2:49 AM

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William Elliot

Posts: 1,553
Registered: 1/8/12
Re: Ultranet
Posted: Nov 16, 2012 2:49 AM
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On Thu, 15 Nov 2012, David C. Ullrich wrote:
> On Thu, 15 Nov 2012 01:28:04 -0800, William Elliot <marsh@panix.com>
> 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.

Any non-trivial examples of ultranets?


Date Subject Author
11/15/12
Read Ultranet
William Elliot
11/15/12
Read Re: Ultranet
David C. Ullrich
11/16/12
Read Re: Ultranet
William Elliot

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