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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
Posted: Jun 30, 2013 10:21 PM
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On 6/30/2013 10:09 AM, 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)
>
> ?
>
>



These are proximity neighborhoods. In the induced
topology they correspond with the usual neighborhood
system which I believe you have described here.

Otherwise not.

http://en.wikipedia.org/wiki/Proximity_space





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