  The concept of regularity in the meta-topological setting of projections in
the double dual of a C*-algebra addresses the interrelations of a projection p
with its closure, for instance in the form that such projections act
identically, in norm, on elements of the C*-algebra. This concept has been
given new actuality with the recent plan of Peligrad and Zsido to find a
meaningful notion of Murray-von Neumann type equivalence among open
projections.
  Although automatic in the commutative case, it has been known since the late
sixties that regularity fails for many projections. The original
investigations, however, did not answer a question such as: "Are all open and
dense projections regular in A, when A is simple?"
  We report here that this and related questions have negative answers. In the
other direction, we supply positive results on regularity of large open
projections.
