  Is a sequence of Riemannian manifolds with positive scalar curvature,
satisfying some conditions to keep the sequence reasonable, compact? What
topology should one use for the convergence and what is the regularity of the
limit space? In this paper we explore these questions by studying the case of a
sequence of Riemannian manifolds which are conformal to the $n$-dimensional
round sphere. We are able to show that the sequence of conformal factors are
compact in several analytic senses and are able to establish $C^0$ convergence
away from a singular set of small volume in a similar fashion as C. Dong. Under
a bound on the total scalar curvature we are able to show that the limit
conformal factor has weak positive scalar curvature in the sense of weakly
solving the conformal positive scalar curvature equation.
