  The Local-to-Global-Principle used in the proof of convexity theorems for
momentum maps has been extracted as a statement of pure topology enriched with
a structure of convexity. We extend this principle to not necessarily closed
maps $f\colon X\ra Y$ where the convexity structure of the target space $Y$
need not be based on a metric. Using a new factorization of $f$, convexity of
the image is proved without local fiber connectedness, and for arbitrary
connected spaces $X$.
