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Topic: Proximity space
Replies: 5   Last Post: Jun 16, 2013 9:39 PM

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David C. Ullrich

Posts: 3,214
Registered: 12/13/04
Re: Proximity space
Posted: Jun 13, 2013 11:16 AM
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On Thu, 13 Jun 2013 02:11:33 -0700, William Elliot <marsh@panix.com>
wrote:

>In Wikipedia's definition of proximity space,
>http://en.wikipedia.org/wiki/Proximity_space
>axioms, definitions and theorems are given which are
>hard to understand because of the strange charactes.
>
>Here's what I've deciphered. How correct is it? A,B,C are subsets of S.
>Writing AnB for "A near to B" or "A close to B", the axioms are:
>
>AnB implies BnA
>AnB implies A not empty
>not emtpy A /\ B implies AnB
>An(B \/ C) implies AnB or AnC


That one is "if and only if", not "implies".

>
>Then theorem for all C, (AnC or B~S\C) implies AnB


That's one of the axioms, not a theorem. Also
it looks to me like your ~ was a typo for n.

_Also_, the way you phrase it is unclear. It looks
like you meant

for all C, [(AnC or BnS\C) implies AnB],

when the axiom is actually

[for all C, (AnC or BnS\C)] implies AnB.


Of course those clumsy brackets should not
be needed if one has a decent command of
the English language. (This is not a complaint
about what's on the Wikipedia page, since
that's written in a more formal-logic style;
just a complaint about your transcription
into standard English.) For example one
might say

If we have AnC or BnS\C for every C
then AnB.

>
>The deffinition A << B for not AnS\B
>and the closure, cl A = { x | {x}nA }.





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