  We show that in an arbitrarily fixed conformal class on a closed manifold,
the upper bound condition of the total scalar curvature is $C^{0}$-closed if
its Yamabe constant is nonpositive. Moreover, we show that if a conformal class
on a closed manifold has positive Yamabe constant, then the intersection of
such conformal class and the space of all Riemannian metrics, whose scalar
curvatures are bounded from below as well as total scalar curvatures are
bounded from above is $C^{0}$-closed in the space of all Riemannian metrics.
