fom
Posts:
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Registered:
12/4/12


Re: Nhood Space
Posted:
Jun 30, 2013 10:21 PM


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

