
Nhood Space
Posted:
Jun 30, 2013 12:39 AM


(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
A << B is taken to mean B is a nhood of A. Thus {x} << A would mean A is a nhood of x.
Additional axioms are separation for all x,y, if x /= y, then {x} << S\y and normality for all A,B, if A << B, then there's some K with A << K << B
Useful theorems are A << B, B subset C implies A << C A subset C, B << C implies A << C
Define the interior of a set A, int A = { x  {x} << A }. Easy theorems are int empty set = empty set; int S = S int A/\B = int A /\ int B; int A subset A A subset B implies int A subset int B.
How would one prove int int A = int A? Since int A subset A, int int A subset int A. So the question actually is how to prove int A subset int int A?
Another question. If for all x in A, {x} << B, is A << B provable?
From the axiom A << B/\C iff A << B and A << C
S\B \/ S\C << S\A iff S\B << S\A and S\C << S\A
Thus for all A,B,C B \/ C << A iff B << A, C << A
Hence if A if finite and for all x in A, {x} << B, then A << B. What if A isn't finite?

