  Recently geometric hypergraphs that can be defined by intersections of
pseudohalfplanes with a finite point set were defined in a purely combinatorial
way. This led to extensions of earlier results about points and halfplanes to
pseudohalfplanes, including polychromatic colorings and discrete Helly-type
theorems about pseudohalfplanes.
  Here we continue this line of research and introduce the notion of convex
sets of such pseudohalfplane hypergraphs. In this context we prove several
results corresponding to classical results about convexity, namely Helly's
Theorem, Carath\'eodory's Theorem, Kirchberger's Theorem, Separation Theorem,
Radon's Theorem and the Cup-Cap Theorem. These results imply the respective
results about pseudoconvex sets in the plane defined using pseudohalfplanes.
  It turns out that most of our results can be also proved using oriented
matroids and topological affine planes (TAPs) but our approach is different
from both of them. Compared to oriented matroids, our theory is based on a
linear ordering of the vertex set which makes our definitions and proofs quite
different and perhaps more elementary. Compared to TAPs, which are continuous
objects, our proofs are purely combinatorial and again quite different in
flavor. Altogether, we believe that our new approach can further our
understanding of these fundamental convexity results.
