We consider the following topological property of a compact Hausdorff space $K$:   for every operator $M$ of multiplication by a real-valued function from $C(K)$, its double commutant coincides with the norm-closed algebra generated by $M$ and the identity operator $I$. If it is the case, we say that $K \in DC$. The main result states that if $K$ is a metrizable connected and locally connected compact space, then $K \in DC$. We also provide examples of metrizable continua not in $DC$ and examples of continua that are not locally or even arc connected but are in $DC$.