reference request – Invariant on C*-algebras-number of closed unbounded derivation it admitted

In working of the unbounded derivation of C*-algebras. I observed the following: For topological manifold $M$, the number of closed, linear independent, unbounded derivation it admitted on $C(M)$ is exactly the dimension of $M$.

Of course this is true for smooth manifold. But I found that it may holds for arbitrary manifold. I try to google it but seems like no positive results. I would like to know if my result is known and well-studied. Thank you in advance.