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

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Peter Percival

Posts: 1,304
Registered: 10/25/10
Re: Nhood Space
Posted: Jul 1, 2013 3:58 AM
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fom wrote:
> 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.


Thank you. I'd been wondering!

> Otherwise not.
>
> http://en.wikipedia.org/wiki/Proximity_space
>
>



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