  We consider the problem of hitting sets online. The hypergraph (i.e.,
range-space consisting of points and ranges) is known in advance, and the
ranges to be stabbed are input one-by-one in an online fashion. The online
algorithm must stab each range upon arrival. An online algorithm may add points
to the hitting set but may not remove already chosen points. The goal is to use
the smallest number of points. The best known competitive ratio for hitting
sets online by Alon et al. \cite{alon2009online} is $O(\log n \cdot \log m)$
for general hypergraphs, where $n$ and $m$ denote the number of points and the
number of ranges, respectively. We consider hypergraphs in which the union of
two intersecting ranges is also a range. Our main result for such hypergraphs
is as follows. The competitive ratio of the online hitting set problem is at
most the unique-max number and at least this number minus one.
