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.
> 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.