Let $M$ be a smooth manifold with non-empty boundary.


Let $g$ be a smooth Riemannian metric on $M$. Is the following true?



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> For every $\epsilon >0$ there exist a Riemannian metric $g\_{\epsilon}$ with non-positive scalar curvature such that $\|g-g\_{\epsilon}\|\_{C^0}<\epsilon$? 
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(Here $\|\cdot\|\_{C^0}$ is the $C^0$ norm w.r.t $g$, but I guess one can use $g\_{\epsilon}$ as well).


I ask whether the metrics of non-positive scalar curvature are dense in the space of metrics.



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> On other hand, I heard that the space of metrics with non-negative scalar curvature is closed in the uniform topology.
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Where can I find references for these two facts?


(I am not sure if the boundary creates problems. I might have heard this about closed manifolds). 


If it matters, I am fine with assuming that $M$ is contractible.



Is every metric uniformly close to a metric with negative scalar curvature?
Jochen Lohkamp answers your first question in "Curvature h-principles", Ann. of Math, vol 142, p 457–498. If $\dim M\ge 3$, then the space of negative scalar curvature metrics is dense in the $C^0$ topology. More remarkably, he even shows this for negative Ricci curvature metrics.


