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Registered:
12/4/12


Re: Nhood Space  proof here
Posted:
Jul 3, 2013 12:02 AM


http://planetmath.org/sites/default/files/texpdf/39037.pdf
at bottom of first page
================================
Look at axiom 5. It is different from what is in Willard or on Wikipedia.
However, it matches the axioms on page 93 of
http://books.google.com/books?id=FdThSVDYTrgC&pg=PA106&lpg=PA106&dq=%22local+proximity+space%22&source=bl&ots=NkGa0_oXAE&sig=CZkYlfEUX0X2x99ZR1xkeTVD3Fk&hl=en&sa=X&ei=VpbTUe7NHIS7ywHnrIHABA&ved=0CDEQ6AEwAQ#v=onepage&q=%22local%20proximity%20space%22&f=false
I believe you will find that there are some uses of underlying set theory that one cannot justify on the basis of the axioms alone.
Based on the axioms alone, the problem I found is that
{y} n cl(A) <> ( ( {y} n A ) \/ ( {y} n cl(A)/A ) )
may be true, but
~( {y} n A ) /\ ( {y} n cl(A)/A )
cannot be shown to be problematic. Perhaps I am just too stupid. In defense of myself, this reminds me of the sequence of derived sets associated with Cantor (and, I believe, descriptive set theory).
In the second link, on page 93 you will find the remark:
"If there exists a point x which is close to both A and B, then A is close to B"
Once again, I think this is from the underlying set theory. But, it also makes the proof trivial.
That is, if
( {y} n {x} ) /\ ( {x} n A )
then
( {y} n A )
(uses symmetry axiom)
and cl cl A subset cl A
================================
To me, the use of the underlying set theory is cheating. When I first read about proximity spaces, I took them to characterize vagueness. Using the underlying set theory in the proofs seems like an assumption of separability regardless of how the nearness relation itself may be defined.
Oh well, I am wrong about many things.



