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

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fom

Posts: 1,969
Registered: 12/4/12
Re: Nhood Space
Posted: Jul 1, 2013 11:48 PM
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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.





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