Upper bound preservation of the total scalar curvature in a conformal class

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.