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If $M$ is a Riemannian manifold that is not compact, wouldis it true that the Sobolev spaces $W^{k,p}(M)$ on $M$ still, $W^{k,p}(M)$, still be separable (for $p < \infty$)?

If $M$ is a Riemannian manifold that is not compact, would Sobolev spaces $W^{k,p}(M)$ on $M$ still be separable (for $p < \infty$)?

If $M$ is a Riemannian manifold that is not compact, is it true that the Sobolev spaces on $M$, $W^{k,p}(M)$, still be separable (for $p < \infty$)?

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Are Sobolev spaces on non-compact manifolds separable?

If $M$ is a Riemannian manifold that is not compact, would Sobolev spaces $W^{k,p}(M)$ on $M$ still be separable (for $p < \infty$)?