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The Betweenness Axioms. For each pair of distinct points a and b there is a subset
 

Summary: The Betweenness Axioms.
For each pair of distinct points a and b there is a subset
s(a, b),
called the segment joining a to b, satisfying (B1)-(B3) below. Whenever a and c are distinct points we
say the point b is between a and c if b s(a, c).
(B1) If a and b are distinct points then
(a) s(a, b) l(a, b);
(b) s(b, a) = s(a, b); and
(c) s(b, a) {a, b} is empty.
(B2)
(a) If a and b are distinct points then s(a, b) is nonempty.
(b) If a, b and c are distinct and {a, b, c} is collinear exactly one of the following holds:
a s(b, c), b s(c, a), c s(a, b).
(c) If a and b are distinct points there is a point c not equal to a or b such that b s(a, c).
Suppose L is a line and a and b are distinct points not lying on L. We say a and b are on the same
side of L if s(a, b) does not meet L and we say a and b are on opposite sides of L if s(a, b) does meet
L. From (B1)(b) we infer that these relations are symmetric in a and b.
The previous axioms imply that a line is at most one dimensional in the sense of topology, whatever
that means. The following axiom says that the set of points is at most two dimensional in the sense of
topology, whatever that means. The continuity axiom, which will follow, will allow us to replace "at most"

  

Source: Allard, William K. - Department of Mathematics, Duke University

 

Collections: Mathematics