  Let A be a C*-algebra and A** its enveloping von Neumann algebra. C. Akemann
suggested a kind of non-commutative topology in which certain projections in
A** play the role of open sets. The adjectives "open", "closed", "compact", and
"relatively compact" all can be applied to projections in A**. Two operator
inequalities were used by Akemann in connection with compactness. Both of these
inequalities are equivalent to compactness for a closed projection in A**, but
only one is equivalent to relative compactness for a general projection. A
third operator inequality, also related to compactness, was used by the author.
It turns out that the study of all three inequalities can be unified by
considering a numerical invariant which is equivalent to the distance of a
projection from the set of relatively compact projections. Since the subject
concerns the relation between a projection and its closure, Tomita's concept of
regularity of projections seems relevant, and some results and examples on
regularity are also given. A few related results on semicontinuity are also
included.
