
Re: Nhood Space
Posted:
Jul 1, 2013 4:02 AM


William Elliot wrote: > On Sun, 30 Jun 2013, William Elliot wrote: >> On Sun, 30 Jun 2013, Peter Percival 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) > > Let's see. Define N(x) = { A  {x} << A } > 1. For all x, {x} << S. > 2. A << B implies A subset B > 3. A << B and B subset C imply A << C > 4. A << B/\C iff A << B, A << C > 5. {x} << S and for all y in S, {y} << S. (Seems trivial.) > > Conversely given N(x), how is A << B to be defined? > {x} << A when A in N(x) ? > A << B when for all a in A, {a} << B ? > > Thus my second question is poignant. > If for all a in A, {a} << B, does A << B? > > If we take A << B to be cl A subset int B and let > A = { 1/n  n in N } and B = [0,1], then not A << B. > > So that rejects my second question leaving me wondering > how to define A << B from the N(x)'s. > > I think your nhood space is more general that my proximal nhood space. > How is an open set defined using N(x)'s?
A set is open if it is a neighbourhood of each of its points.
> U open when for all x in U, some V in N(x) with V subset U? > Thus empty set is open. > > It seems forthwith, that the collection of open sets defines > a topology and every topology gives a nhood space, which if > derived from N(x)'s, will give the same nhood space in return. > > It seems proximal nhood spaces don't generate every topology > and accordingly less general. They were, in fact, not intended > to describe topological spaces but merely to generalize uniform > spaces. >
 I think I am an Elephant, Behind another Elephant Behind /another/ Elephant who isn't really there.... A.A. Milne

