Singular precalibers for topological products

In this paper we prove that if $\kappa$ is a singular cardinal with uncountable cofinality, then every power of a given topological space with precaliber $\kappa$ has precaliber $\kappa$ as well. Furthermore, if $\{X_\alpha : \alpha<\lambda\}$ is a family of topological spaces with precaliber $\kappa$ and $\lambda<cf(\kappa)$, then $\kappa$ is a precaliber for the topological product $\prod\{X_\alpha : \alpha<\lambda\}$.