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Re: Proximity space
Posted:
Jun 13, 2013 11:16 AM


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 formallogic 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 }.



