Classes of compact spaces

Given a class $\mathcal{C}$ of compact Hausdorff spaces which is closed under countable products and continuous images. Let $\kappa>\omega$ be a cardinal number. Consider the class $\mathcal{C}^\prime$ consisting of compact spaces produced by taking products of less than $\kappa$ spaces from the class $\mathcal{C}$. Must the class $\mathcal{C}^\prime$ be closed under continuous images?

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Any concrete examples where the hypothesis holds? –  tomasz Jun 25 '12 at 13:25