fom
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Registered:
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Re: Nhood Space
Posted:
Jul 1, 2013 11:48 PM


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.
Then for cl(A) one has
x near A implies that for every entourage (surrounding) D there exists some t in S such that
(x,t) implies t in D[{x}]
and
(y,t) implies t in D[A]
So, D[A] fixes the values of t that determine whether any given x is near.
So, this should mean that
D[A] = D[cl(A)]
so
cl(A) = cl(cl(A))
Anyway, its late. Maybe I screwed this up.
But, it should give you a different way to think about the problem.

