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

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fom

Posts: 1,968
Registered: 12/4/12
Re: Nhood Space -- proof here
Posted: Jul 3, 2013 12:02 AM
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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.













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