On countable \(\sigma\)-product spaces (Q1801724)

As for countable \(\sigma\)-products the author proves that if every finite subproduct of \(X = \sigma \{X_i : i \in \omega\}\) is normal, then \(X\) is normal if and only if \(X\) is countably paracompact. However it remains open whether the above result is true or not for arbitrary \(\sigma\)- products. As a partial result it is shown that if every finite subproduct of \(X = \sigma\{X_\alpha : \alpha \in \lambda\}\) is normal and \(X\) is \(\lambda\)-paracompact, then \(X\) is normal.