William Elliot wrote: > On Mon, 1 Jul 2013, Peter Percival wrote: >> 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 > > LIkely not for the reasons I previously indicated. > >> b) can N be defined in terms of <<, and can my axioms be deduced as theorems >> from William's? > > Yes, I showed you how. > >> And of course everyone knew that's what I meant. > > Popular Presumption. > >> I wrote of my axioms just for reasons of euphony, I think they are >> Hausdorff's. Again, someone more knowledgeable than I will know. > > Here's from the web. Are these equivalent to yours? > > The axioms formulated by Hausdorff (1919) for his concept of a > topological space. These axioms describe the properties satisfied > by subsets of elements x in a neighborhood set E of x . > > 1. There corresponds to each point x at least one neighborhood > U(x) , and each neighborhood U(x) contains the point x . > > 2. If U(x) and V(x) are two neighborhoods of the same point x , > there must exist a neighborhood W(x) that is a subset of both. > > 3. If the point y lies in U(x) , there must exist a neighborhood > U(y) that is a subset of U(x) . > > 4. For two different points x and y , there are two corresponding > neighborhoods U(x) and U(y) with no points in common.
Mine doesn't imply the separation axiom 4.
-- I think I am an Elephant, Behind another Elephant Behind /another/ Elephant who isn't really there.... A.A. Milne