  The notion of stationary reflection is one of the most important notions of
combinatorial set theory. We investigate weak reflection, which is, as the name
suggests, a weak version of stationary reflection. This sort of reflection was
introduced in [DjSh:545] (math.LO/9601219), where it was shown that weak
reflection has applications to various guessing principles, in the sense that
if there is no weak reflection, than a guessing principle holds, and an
application dealing with the saturation of normal filters. Further
investigations of weak reflection were carried in [CDSh:571] (math.LO/9504221)
and [CuSh:596] (math.LO/9703219). While various ZFC restrictions on the one
hand, and independence results on the other, were discovered about the weak
reflection, the question of the relative consistency of the existence of a
regular cardinal kappa such that the first cardinal weakly reflecting at kappa
is a successor of singular, remained open. This paper answers that question by
proving that (modulo large cardinal assumptions close to 2-hugeness) that there
indeed can be such a cardinal kappa.
