
Re: Nhood Space
Posted:
Jul 1, 2013 12:03 PM


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

