Abstract

We investigate in ZF conditions that are necessary and sufficient for countable products ∏m∈ℕXm of finite Hausdorff spaces Xm resp. Hausdorff spaces Xm with at most n points to be compact resp. Baire. Typica results: Countable products of finite Hausdorff spaces are compact if and only if countable products of non-empty finite sets are non-empty. Countable products of discrete spaces with at most n + 1 points are compact if and only if countable products of non-empty sets with at most n points are non-empty