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

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Peter Percival

Posts: 1,254
Registered: 10/25/10
Re: Nhood Space
Posted: Jul 1, 2013 12:03 PM
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Peter Percival wrote:
> William Elliot wrote:
>> (S,<<) is a nhood space when << is a binary relation for P(S) and
>> for all A,B,C subset S
>> empty set << A << S
>> A << B implies A subset B
>> A << B implies S\B << S\A
>> A << B/\C iff A << B and A << C

>
>
> Is this the same as neighbourhood space defined as follows.
>
> (S, N), S a set, N a map S -> PPS (P for power set) and
>
> i) x in S => N(x) =/= 0
>
> ii) x in S, M in N(x) => x in M
>
> iii) x in S, M in N(x) => (L superset M => L in N(x)
>
> iv) x in S, L, M in N(x) => L intersect M in N(x)
>
> v) x in S, M in N(x) => exists L in N(x) s.t.
> L subset M and, forall y in L, L in N(y)
>
> ?


What I meant when asking are they the same was:
a) can << be defined in terms of N, and can William's axioms be deduced
as theorems from mine; and
b) can N be defined in terms of <<, and can my axioms be deduced as
theorems from William's?

And of course everyone knew that's what I meant.

But there is probably some official definition of "the same as" in terms
of natural transformations. Someone more knowledgeable than I will know.

I wrote of my axioms just for reasons of euphony, I think they are
Hausdorff's. Again, someone more knowledgeable than I will know.

--
I think I am an Elephant,
Behind another Elephant
Behind /another/ Elephant who isn't really there....
A.A. Milne



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