  A coordinate cone in R^n is an intersection of some coordinate hyperplanes
and open coordinate half-spaces. A semi-monotone set is a defnable in an
o-minimal structure over the reals, open bounded subset of R^n such that its
intersection with any translation of any coordinate cone is connected. This can
be viewed as a generalization of the convexity property. Semi-monotone sets
have a number of interesting geometric and combinatorial properties. The main
result of the paper is that every semi-monotone set is a topological regular
cell.
