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

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William Elliot

Posts: 2,637
Registered: 1/8/12
Re: Nhood Space -- proof here
Posted: Jul 3, 2013 10:43 PM
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On Tue, 2 Jul 2013, fom wrote:

> at bottom of first page

Great find!

> Look at axiom 5. It is different from what
> is in Willard or on Wikipedia.

Equivalent form of normality.

> However, it matches the axioms on page 93 of

Not possible to view because I've imediately "reached the viewing limit".

Now that you've given proof of cl cl A = cl A, I see the roll normality plays
in the proof and use that to give a simple and direct proof int int A = int A
= { x | {x} << A } without any mind twisting double negatives.
But first noted that A << B implies A subset int B.

Since int A subset A, int int A subset int A.
To prove int A subset int int A, assume x in int A.
Thus {x} << A and by normality there's some K with {x} << K, K << A.
Hence by the note above; {x} subset int K, K subset int A.
Consequently x in int K subset int int A, QED.

To continue with the insprition you've proveded note in passing that
int S\A = S\cl A. Thus the following follows from A << B:
S\B << S\A; S\B subset int S\A = S\cl A; cl A subset B

Using that result and normality with A << B, there's some
K with A << K << B. Whence cl A subset K subset int B.

In conclusion, A << B implies cl A subset int B
(in fact some K with cl A subset int K, cl K subset int B)
within the topology { int A | A subset S } of the
proximity nhood space (S,<<).

That however doesn't imply normal p-nhood spaces induce normal topologies,
as tempting it is to think it would.

> 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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