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Topic:
Nhood Space
Replies:
24
Last Post:
Jul 3, 2013 10:43 PM



fom
Posts:
1,968
Registered:
12/4/12


Re: Nhood Space
Posted:
Jul 2, 2013 6:23 AM


On 7/2/2013 12:19 AM, William Elliot wrote: > On Mon, 1 Jul 2013, fom wrote: >> On 6/30/2013 9:03 PM, William Elliot wrote: > >>> Nhood spaces are the DeMorgan like duals of proximity spaces. >>> Williard gives a closure operator cl, for a proximity space >>> and leaves it as an exercise to show cl is a closure operator. >>> The part I'm having trouble with is proving cl cl A = cl A, >>> which in the dual nhood space is int int A = int A. >> >> Look at the example in Willard concerning >> the diagonal uniformities. And then look >> at the definition for neighborhoods in a >> uniformity in the previous sections. > > Arrg. It's hard to follow what your thinking and how it apllies to the > problem at hand.
The problem at hand seems to be how to get to the result.
What I am thinking is that a collection of terms that refer to the same object (a partition on a class of names) are all near to one another.
Two subsets from different partitions may be near to one another if they have some overlapping terms.
A separated uniformity just converges to the diagonal, <x,x>
So, this "nearness" in proximity spaces seems to be related to subsets associated with the lattice of equivalence relations.
I'll come back to it after work.
Found this, though. It may have more in it than just an axiom list.
http://matwbn.icm.edu.pl/ksiazki/fm/fm47/fm47112.pdf
Another way to think about what I am talking about is to think of Cantor's intersection theorem with diameters converging to some nonzero epsilon.
Take these neighborhoods as the "subsets without proper subsets" which constitute "points" as mentioned in the pdf.
Also, there had been the link in the Wikipedia page that mentioned Cantorian derived sets. That is why one of the axioms looks like the betweeness condition of the rationals.



