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Assuming Jensen's diamond principle ($\diamondsuit$) we construct for everynatural number $n>0$ a compact Hausdorff space $K$ such that whenever theBanach spaces $C(K)$ and $C(L)$ are isomorphic for some compact Hausdorff $L$,then the covering dimension of $L$ is equal to $n$. The constructed space $K$is separable and connected, and the Banach space $C(K)$ has few operators i.e.every bounded linear operator $T:C(K)\rightarrow C(K)$ is of the form$T(f)=fg+S(f)$, where $g\in C(K)$ and $S$ is weakly compact.

A Banach space $C(K)$ reading the dimension of $K$