On Countable Products of Finite Hausdorff Spaces

We investigate in ZF (i.e., Zermelo‐Fraenke set theory without the axiom of choice) conditions that are necessary and sufficient for countable products ∏_m∈ℕX_m of (a) finite Hausdorff spaces X_m resp. (b) Hausdorff spaces X_m with at most n points to be compact resp. Baire. Typica results: (i) Countable products of finite Hausdorff spaces are compact (resp. Baire) if and only if countable products of non‐empty finite sets are non‐empty. (ii) Countable products of discrete spaces with at most n + 1 points are compact (resp. Baire) if and only if countable products of non‐empty sets with at most n points are non‐empty.