A cardinal $\kappa$ is product-regular if, for all families of sets $(X_i)_{i \in I}$, if $\vert I \vert \lt \kappa$ and for all elements $i \in I$$\vert X_i \vert \lt \kappa$, then the indexed product

$\left| \prod_{i \in I} X_i \right| \lt \kappa$

Properties

Every inaccessible cardinal is product-regular. Every uncountable product-regular cardinal is an inaccessible cardinal, but the finite cardinals $0$ representing the empty set and the countable cardinal $\aleph_0$ representing the natural numbers are product-regular cardinals which are not inaccessible cardinals.