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Re: Motion P1788/M0013.04 - Comparisons

Nate Hayes wrote:

John Pryce wrote:

On 18 Sep 2010, at 23:15, Nate Hayes wrote:

I speak against this.
Ulrich's interior is better.
Note that the topological interior, i.e., "proper subset," is already expressed efficiently in terms of Ulrich's relations for intervals A,B: 
  ( A \subset B ) and not ( A == B )

Doesn't that make [2,3] interior to [1,3]?


I don't see Entire should be interior to Entire.
Well, it seems weird to me too, but there it is. You're an expert on quantified statements. Isn't it inescapable from the definition "B is a neighbourhood of each a in A" (eqn1)?
As you are so fond of quoting from George Corliss: if it even seems weird to you -- a seasoned mathemetician -- then "God help the casual user!"
The thing about definitions grounded in standard theory (and this theory has been around for roughly 100 years) is that, compared with ad-hoc definitions, you KNOW they can't lead to inconsistencies -- assuming math itself is consistent.
This is a reason I think P1788 might want to investigate sticking to compact intervals, instead.... which if you remember was one of my first choices.
Unbounded intervals are essential for applications to general global optimization and nonlinear systems solving, where for complex models 
one often doesn't know in advance bounds on all variables.

On the other hand, the casual user will hardly come into a situation
where the interior is needed. Whoever understands the proof of existence
tests involving theinterior condition also understands the elementary topological concepts. 


Arnold Neumaier