```Date: Jun 30, 2013 12:39 AM
Author: William Elliot
Subject: Nhood Space

(S,<<) is a nhood space when << is a binary relation for P(S) andfor 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 << CA << 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\yand normality	for all A,B, if A << B, then there's some K with		A << K << BUseful theorems areA << B, B subset C implies A << CA subset C, B << C implies A << CDefine 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\AThus for all A,B,C	B \/ C << A iff B << A, C << AHence if A if finite and for all x in A, {x} << B, then A << B.What if A isn't finite?
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