Real rank of $C_0(R)$

Question

I was trying to calculate Real rank of $C_0(\mathbb{R} \otimes \mathbb{K}(l^2))$. $\mathbb{K}$, compact operators algebra, is inductive limit of matrix algebras. So, $RR(C_0(\mathbb{R} \otimes \mathbb{K}(l^2)) = lim_{\rightarrow}RR (C_0(\mathbb{R} \otimes \mathbb{M_n})) = C_0(\mathbb{R})$. But what the real rank of $C_0(\mathbb{R})$?

Answer 1

I think I found answer: If $X$ is a locally compact Hausdorff space, then $RR(C_0(X)) = dim(X)$, where $dim$ is the Lebesgue covering dimension of $X$.