| dc.description.abstract | Axiomatic convexity space, introduced by Kay and Womble [22] ,
will be the main topic discussed in this thesis.
An axiomatic convexity space (X,C), which is domain finite
and has regular straight segments, is called a basic convexity
space, A weak complete basic convexity space is a basic convexity
space which is complete and has C-isomorphic property. If in
addition, it is join-hull commutative then it is called (strong)
complete basic convexity space.
The main results presented are: a generalized line space is
a weak complete basic convexity space, a complete basic convexity
space is equivalent to a line space; and a complete basic convexity
space whose dimension is greater than two or desarguesian and of
dimension two, is a linearly open convex subset of a real affine
space.
Finally, we develop a linearization theory by following an
approach given by Bennett [3]. A basic convexity space whose
dimension is greater than two, which is join-hull commutative
and has a parallelism property, is an affine space. It can be
made into a vector space over an ordered division ring and the
members of C are precisely the convex subsets of the vector
space. | |
